GEORGE L. HOSMER
Assist ant Professor of Civil Engineering, Massachusetts
Institute of Technology
JOHN WILEY & SONS
LONDON: CHAPMAN & HALL, LIMITED
GEORGE L. HOSMER
F. H. CILSON COMPANY
THE purpose of this volume is to furnish a text in Practical
Astronomy especially adapted to the needs of civil-engineering
students who can devote but little time to the subject, and who
are not likely to take up advanced study of Astronomy. The
text deals chiefly with the class of observations which can be
made with surveying instruments, the methods applicable to
astronomical and geodetic instruments being treated but briefly.
It has been the author's intention to produce a book which is
intermediate between the text-book written for the student of
Astronomy or Geodesy and the short chapter on the subject
generally given in text-books on Surveying. The subject has
therefore been treated from the standpoint of the engineer, who
is interested chiefly in obtaining results, and those refinements
have been omitted which are beyond the requirements of the
work which can be performed with the engineer's transit. This
has led to the introduction of some rather crude mathematical
processes, but it is hoped that these are presented in such a way
as to aid the student in gaining a clearer conception of the prob-
lem without conveying wrong notions as to when such short-cut
methods can properly be applied. The elementary principles
have been treated rather elaborately but with a view to making
these principles clear rather than to the introduction of refine-
ments. Much space has been devoted to the Measurement of
Time because this subject seems to cause the student more
difficulty than any other branch of Practical Astronomy. The
attempt has been made to arrange the text so that it will be a
convenient reference book for the engineer who is doing field
For convenience in arranging a shorter course those subjects
which are most elementary are printed in large type. The mat-
ter printed in smaller type may be included in a longer course
and will be found convenient for reference in field practice, par-
ticularly that contained in Chapters X to XIII.
The author desires to acknowledge his indebtedness to those
who have assisted in the preparation of this book, especially to
Professor A. G. Robbins and Mr. J. W. Howard of the Massa-
chusetts Institute of Technology and to Mr. F. C. Starr of the
George Washington University for valuable suggestions and crit-
icisms of the manuscript.
G. L. H.
BOSTON, June, 1910.
TABLE OF CONTENTS
THE CELESTIAL SPHERE REAL AND APPARENT MOTIONS
1. Practical Astronomy i
2. The Celestial Sphere i
3. Apparent Motion of the Sphere 3
4. The Motions of the Planets 3
5. Meaning of Terms East and West 6
6. The Earth's Orbital Motion The Seasons 7
7. The Sun's Apparent Position at Different Seasons 9
8. Precession and Nutation 10
9. Aberration of Light 12
DEFINITIONS POINTS AND CIRCLES OF REFERENCE
10. Definitions 14
Vertical Line Zenith Nadir Horizon Vertical Circles
Almucantars Poles Equator Hour Circles Par-
allels of Declination Meridian Prime Vertical Eclip-
tic Equinoxes Solstices Colures.
SYSTEMS OF COORDINATES ON THE SPHERE
11. Spherical Coordinates 18
12. The Horizon System 19
13. The Equator Systems 19
15. Coordinates of the Observer 22
16. Relation between the Two Systems of Coordinates 23
VI TABLE OF CONTENTS
RELATION BETWEEN COORDINATES
17. Relation between Altitude of Pole and Latitude of Observer. ... 27
1 8. Relation between Latitude of Observer and the Declination and
Altitude of a Star on the Meridian 30
19. The Astronomical Triangle 31
20. Relation between Right Ascension and Hour Angle 36
MEASUREMENT OF TIME
21. The Earth's Rotation 39
22. Transit or Culmination 39
23. Sidereal Day 39
24. Sidereal Time 40
25. Solar Day 40
26. Solar Time ' 40
27. Equation of Time 41
28. Conversion of Apparent Time into Mean Time and vice versa ... 43
29. Astronomical and Civil Time 44
30. Relation between Longitude and Time 45
31. Relation between Sidereal Tune, Right Ascension and Hour Angle
of any Point at a. Given Instant 48
32. Star on the Meridian 49
33. Relation between Mean Solar and Sidereal Intervals of Time. ... 49
34. Relation between Sidereal and Mean Time at any Instant 52
35. Standard Time 56
36. The Date Line 58
37. The Calendar 59
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC STAR
38. The Ephemeris 62
39. Star Catalogues 69
40. Interpolation 69
TABLE OF CONTENTS Vll
THE EARTH'S FIGURE CORRECTIONS TO OBSERVED ALTITUDES
41. The Earth's Figure 72
42. Parallax 73
43. Refraction 76
44. Semidiameters 78
45- Dip 79
46. Sequence of Corrections 80
DESCRIPTION OF INSTRUMENTS OBSERVING
47. The Engineer's Transit 82
48. Elimination of Errors 83
49. Attachments to the Engineer's Transit Reflector 86
50. Prismatic Eyepiece 87
51. Sun Glass 87
52. The Portable Astronomical Transit 87
53. The Sextant 88
54. Artificial Horizon 91
55. Chronometer 92
56. Chronograph. 93
57. The Zenith Telescope 94
58. Suggestions about Observing 95
59. The Constellations 98
60. Method of Naming Stars 98
61. Magnitudes 99
62. Constellations near the Pole 99
63. Constellations near the Equator 100
64. The Planets 102
OBSERVATIONS FOR LATITUDE
65. Latitude by a Circumpolar Star at Culmination 103
66. Latitude by Altitude of the Sun at Noon 105
67. Latitude by the Meridian Altitude of a Southern Star 107
Vlll TABLE OF CONTENTS
68. Latitude by Altitudes Near the Meridian 108
69. Latitude by Polaris when the Time is Known no
70. Precise Latitude Determinations Talcott's Method 112
OBSERVATIONS FOR DETERMINING THE TIME
71. Observations for Local Time 114
72. Time by Transit of a Star 114
73. Observations with Astronomical Transit 117
74. Selecting Stars for Transit Observations 117
75. Time by Transit of the Sun 119
76. Time by Altitude of the Sun 120
77. Time by Altitude of a Star 123
78. Tune by Transit of Star over Vertical Circle through Polaris. ... 124
79. Time by Equal Altitudes of a Star 127
80. Time by Two Stars at Equal Altitudes 128
83. Rating a Watch by Transit of a Star over a Range 135
84. Time Service 136
OBSERVATIONS FOR LONGITUDE
85. Methods of Measuring Longitude 139
86. Longitude by Transportation of Timepiece 139
87. Longitude by the Electric Telegraph 140
88. Longitude by Transit of the Moon 141
OBSERVATIONS FOR AZIMUTH
89. Determination of Azimuth 146
90. Azimuth Mark 146
91. Azimuth by Polaris at Elongation 147
92. Observations near Elongation 149
93. Azimuth by an Altitude of the Sun 151
94. Azimuth by an Altitude of a Star !55
95. Azimuth Observation on a Circumpolar Star at any Hour Angle . 155
96. The Curvature Correction 158
TABLE OF CONTENTS ix
97. The Level Correction 158
98. Diurnal Aberration 158
99. Meridian by Polaris at Culmination 161
100. Azimuth by Equal Altitudes of a Star 164
101. Observation for Meridian by Equal Altitudes of the Sun 165
102. Observation of the Sun near Noon 166
103. Combining Observations 167
104. Observations at Sea 170
Determination of Latitude at Sea:
105. Latitude by Noon Altitude of the Sun 170
106. Latitude by Ex-Meridian Altitudes 171
Determination of Longitude at Sea:
107. Longitude by the Greenwich Time and the Sun's Altitude. ... 172
108. Longitude by the Lunar Distance 172
109. Azimuth of the Sun at a Given Time 174
no. Azimuth of the Sun by Altitude and Time 175
in. Sumner's Method of Determining a Ship's Position 175
112. Position by Computation 178
I. MEAN REFRACTION 184
II. CONVERSION OF SIDEREAL TO SOLAR TIME 185
III. CONVERSION OF SOLAR TO SIDEREAL TIME 186
IV. (A) SUN'S PARALLAX (B) SUN'S SEMIDIAMETER (C) DIP OF
V. TIMES OF CULMINATION AND ELONGATION OF POLARIS 188
VI. CORRECTION TO OBSERVE ALTITUDE OF POLARIS 189
VII. VALUES OF FACTOR 112.5 X 3600 X sin i" X tan Z e 190
GREEK ALPHABET 190
LIST OF ABBREVIATIONS 191
THE CELESTIAL SPHERE REAL AND APPARENT
1. Practical Astronomy.
Practical Astronomy treats of the theory and use of astro-
nomical instruments and the methods of computing the results
obtained by observation. The part of the subject which is of
especial importance to the surveyor is that which deals with the
methods of locating points on the earth's surface and of ori-
enting the lines of a survey, and includes the determination of
(i) latitude, (2) time, (3) longitude, and (4) azimuth. In solving
these problems the observer makes measurements of the direc-
tions of the sun, moon, stars, and other heavenly bodies; he is
not concerned with the distances of these objects, with their
actual motions in space, nor with their physical characteristics,
but simply regards them as a number of visible objects of known
positions from which he can make his measurements.
2. The Celestial Sphere.
Since it is only the directions of these objects that are required
in practical astronomy, it is found convenient to regard all
heavenly bodies as being situated on the surface of a sphere
whose radius is infinite and whose centre is at the .eye of the
observer. The apparent position of any object on the sphere is
found by imagining a line drawn from the eye to the object, and
prolonging it until it pierces the sphere. For example, the
apparent position of Si on the sphere (Fig. i) is at Si, which is
supposed to be at an infinite distance from C; the position of
S z is Sz, etc. By means of this imaginary sphere all problems
2 PRACTICAL ASTRONOMY
involving the angular distances between points, and angles
between planes through the centre of the sphere, may readily be
solved by applying the formulae of spherical trigonometry.
This device is not only convenient for mathematical purposes,
but it is perfectly consistent with what we see, because all celestial
objects are so far away that they appear to the eye to be at the
same distance, and consequently on the surface of a great sphere.
FIG. i. APPARENT POSITIONS ON THE SPHERE
From the definition it will be apparent that each observer sees
a different celestial sphere, but this causes no actual inconve-
nience, for distances between points on the earth's surface are so
short when compared with astronomical distances that they are
practically zero except for the nearer bodies in the solar system.
This may be better understood from the statement that if the
entire solar system be represented as occupying a field one mile
in diameter the nearest star would be about 5000 miles away on
the same scale; furthermore the earth's diameter is but a minute
fraction of the distance across the solar system, thematic being
about 8000 miles to 5,60x3,000,000 miles,* or one 7oo,oooth part
of this distance.
* The diameter of Neptune's orbit.
THE CELESTIAL SPHERE 3
Since the radius of the celestial sphere is infinite, all of the
lines in a system of parallels will pierce the sphere in the same
point, and parallel planes at any finite distance apart will cut
the sphere in the same great circle. This must be kept constantly
in mind when representing the sphere by means of a sketch, in
which minute errors will necessarily appear to be very large.
The student should become accustomed to thinking of the
appearance of the sphere both from the inside and from an out-
side point of view. It is usually easier to understand the spheri-
cal problems by studying a small globe, but when celestial
objects are actually observed they are necessarily seen from a
point inside the sphere.
3. Apparent Motion of the Celestial Sphere.
If a person watches the stars for several hours he will see that
they appear to rise in the east and to set in the west, and that
their paths are arcs of circles. By facing to the north (in the
northern hemisphere) it will be found that the circles are smaller
and all appear to be concentric about a certain point in the sky
called the pole ; if a star were exactly at this point it would have
no apparent motion. In other words, the whole celestial sphere
appears to be rotating about an axis. This apparent rotation
is found to be due simply to the actual rotation of the earth
about its axis (from west to east) in the opposite direction to
that in which the stars appear to move.*
4. Motions of the Planets.
If an observer were to view the solar system from a point far
outside, looking from the north toward the south, he would see
that all of the planets (including the earth) revolve about the
sun in elliptical orbits which are nearly circular, the direction
of the motion being counter-clockwise or left-handed rotation.
* This apparent rotation may be easily demonstrated by taking a photo-
graph of the stars near the pole, exposing the plate for several hours. The
result is a series of concentric arcs all subtending the same angle. If the
camera is pointed southward and high enough to photograph stars near the
equator the star trails appear as straight lines.
He would also see that the earth rotates on its axis, once
per day, in a counter-clockwise direction. The moon revolves
around the earth in an orbit which is not so nearly circular,
but the motion is in the same (left-handed) direction. The
FIG. 2. DIAGRAM OF THE SOLAR SYSTEM WITHIN THE ORBIT OF SATURN
apparent motions resulting from these actual motions are as
follows: The whole celestial sphere, carrying with it all the
stars, sun, moon, and planets, appears to rotate about the earth's
axis once per day in a clockwise (right-handed) direction. The
stars change their positions so slowly that they appear to be fixed
in position on the sphere, whereas all objects within the solar
system rapidly change their apparent positions among the stars.
For this reason the stars are called fixed stars to distinguish them
from the planets; the latter, while closely resembling the stars
THE CELESTIAL SPHERE 5
in appearance, are really of an entirely different character. The
sun appears to move slowly eastward among the stars at the rate
of about i per day, and to make one revolution around the earth
FIG. 3a. SUN'S APPARENT POSITION AT GREENWICH NOON ON MAY 22, 23,
AND 24, 1910
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-
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
FIG. 4. APPARENT PATH OF JUPITER FROM OCT., 1909 TO OCT., 1910.
the sun and an apparent motion caused by the earth's revolution
around the sun; the planets consequently appear at certain
times to move westward (i.e., backward), or to retrograde.
Fig. 4 shows the loop in the apparent path of the planet Jupiter
caused by the earth's motion around the sun. It will be seen
that the apparent motion of the planet was direct except from
January to June, 1910, when it had a retrograde motion.
5. Meaning of Terms East and West.
In astronomy the terms " east " and " west " cannot be taken
to mean the same as they do when dealing with directions in one
THE CELESTIAL SPHERE
plane. In plane surveying " east " and " west " may be con-
sidered to mean the directions perpendicular to the meridian
line. If a person at Greenwich
(England) and another person at
the 1 80 meridian should both
point due east, they would actu-
ally be pointing to opposite points
of the sky. In Fig. 5 all four of
the arrows are pointing east at the
places shown. It will be seen from
this figure that the terms " east "
and " west " must therefore be
taken to mean directions of ro-
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
FIG. 6. THE EARTH'S ORBITAL MOTION
of an ellipse, the sun being at one of the foci. Since the earth is
maintained in its position by the force of gravitation, it moves, as
a consequence, at such a speecl in each part of its path that the
line joining the earth and sun moves over equal areas in equal
times. In Fig. 6 all of the shaded areas are equal and the arcs
aa'j bb', cc' represent the distances passed over in the same num-
ber of days.*
The axis of rotation of the earth is inclined to the plane of the
orbit at an angle of about 66^, that is, the plane of the earth's
equator is inclined at an angle of about 23^ to the plane of the
orbit. This latter angle is known as the obliquity of the ecliptic.
(See Chapter II.) The direction of the earth's axis of rotation
is nearly constant and it therefore points nearly to the same
place in the sky year after year.
The changes in the seasons are a direct result of the inclination
of the axis and of the fact that the axis remains nearly parallel
Perihelion (Dec. 31)
FIG. 7. THE SEASONS
to itself. When the earth is in that part of the orbit where the
northern end of the axis is pointed away from the sun (Fig. 7)
it is winter in the northern hemisphere. The sun appears to be
* The eccentricity of the ellipse shown in Fig. 6 is exaggerated for the sake
of clearness ; the earth's orbit is in reality much more nearly circular, the
variation in the earth's distance from the sun being only about three per cent.
THE CELESTIAL SPHERE 9
farthest south about Dec. 21, and at this time the days are
shortest and the nights are longest. When the earth is in this
position, a plane through the axis and perpendicular to the plane
of the orbit will pass through the sun. About ten days later the
earth passes the end of the major axis of the ellipse and is at its
point of nearest approach to the sun, or perihelion. Although
the earth is really nearer to the sun in winter than in summer,
this has but a small effect upon the seasons; the chief reasons
why it is colder in winter are that the day is shorter and the
rays of sunlight strike the surface of the ground more obliquely.
The sun appears to be farthest north about June 22, at which
time summer begins in the northern hemisphere and the days are
longest and the nights shortest. When the earth passes the
other end of the major axis of the ellipse it is farthest from the
sun, or at aphelion. On March 21 the sun is in the plane of
the earth's equator and day and night are of equal length at all
places on the earth (Fig. 7). On Sept. 22 the sun is again in
the plane of the equator and day and night are everywhere
equal. These two times are called the equinoxes (vernal and
autumnal), and the points in the sky where the sun's centre ap-
pears to be at these two dates are called the equinoctial points,
or more commonly the equinoxes.
7. The Sun's Apparent Position at Different Seasons.
The apparent positions of the sun on the celestial sphere
corresponding to these different positions of the earth .are shown
in Fig. 8. As a result of the sun's apparent eastward motion
from day to day along a path which is inclined to the equator,
the angular distance of the sun from the equator is continually
changing. Half of the year it is north of the equator and half of
the year it is south. On June 22 the sun is in its most northerly
position and is visible more than half the day to a person in the
northern hemisphere (/, Fig. 8). On Dec. 21 it is farthest south
of the equator and is visible less than half the day (Z>, Fig. 8).
In between these two extremes it moves back and forth across
the equator, passing it about March 21 and Sept. 22 each year.
The apparent motion of the sun is therefore a helical motion
about the axis, that is, the sun, instead of following the path
which would be followed by a fixed star, gradually increases or
decreases its angular distance from the pole at the same time
that it revolves once a day around the earth. The sun's motion
eastward on the celestial sphere, due to the earth's orbital motion,
FIG. 8. SUN'S APPARENT POSITION AT DIFFERENT SEASONS
is not noticed until the sun's position is carefully observed with
reference to the stars. If a record is kept for a year showing
which constellations are visible in the east soon after sunset,
it will be found that these change from month to month, and at
the end of a year the one first seen will again appear in the east,
showing that the sun has apparently made the circuit of the
heavens in an eastward direction
8. Precession and Nutation.
While the direction of the earth's rotation axis is so nearly
constant that no change is observed during short periods of
time, there is hi reality a very slow progressive change in its
direction. This change is due to the fact that the earth is not
quite spherical in form but is spheroidal, and there is in conse-
quence a ring of matter around the equator upon which the
sun and the moon exert a force of attraction which tends to pull
the plane of the equator into coincidence with the plane of the
orbit. But since the earth is rotating with a high velocity and
THE CELESTIAL SPHERE
resists this attraction, the actual effect is not to permanently
change the inclination of the equator to the orbit, but first to
cause the earth's axis to describe a cone about an axis per-
pendicular to the orbit, and second to cause the inclination of
the axis to go through certain periodic changes (see Fig. 9). The
movement of the axis in a conical surface causes the line of
intersection of the equator and the plane of the orbit to revolve
slowly westward, the pole itself always moving directly toward
the vernal equinox. This causes the equinoctial points to move
westward in the sky, and hence the sun crosses the equator each
spring earlier than it would otherwise; this is known as the
FIG. 9. PRECESSION OF THE EQUINOXES
precession of the equinoxes. In Fig. 9 the pole occupies suc-
cessively the positions /, 2 and J, which causes the point V to
move to points i, 2 and 3- This motion is but 50". 2 per year,
and it therefore requires about 25,800 years for the pole to make
one complete revolution. The force causing the precession is
not quite constant, and the motion of the equinoctial points is
therefore not perfectly uniform but has a small periodic varia-
tion. In addition to this periodic change in the rate of the
precession there is also a slight periodic change in the obliquity,
called Nutation. The maximum value of the nutation is about
9"; the period is about 19 years. The phenomenon of preces-
sion is clearly illustrated by means of the apparatus called the
gyroscope. As a result of the precessional movement of the
axis all of the stars gradually change their positions with refer-
ence to the plane of the equator and the position of the equinox.
The stars themselves have but a very slight angular motion,
this apparent change in position being due almost entirely to the
change in the positions of the circles of reference.
9. Aberration of Light.
Another apparent displacement of the stars due to the earth's
motion is what is known as aberration. On account of the
rapid motion of the earth through space, the direction in which
a star is seen by an observer is a result of the combined velocities
of the observer and of light from the star. The star always
appears to be slightly displaced in the direction in which the
observer is actually moving. In Fig. 10, if light moves from C
to B in the same length of time that the observer moves from
A to B, then C would appear to be in the direction AC. This
may be more clearly understood by using the familiar illustra-
tion of the falling raindrop. If a raindrop is falling vertically,
CB, Fig. n, and while it is falling a person moves from A to B,
then, considering only the two motions, it appears to the person'
that the raindrop has moved toward him in the direction CA.
If a tube is to be held in such a way that the raindrop shall pass
through it without touching the sides, it must be held at the
THE CELESTIAL SPHERE 13
inclination of AC. The apparent displacement of a star due
to the observer's motion is similar to the change in the apparent
direction of the raindrop.
There are two kinds of aberration, annual and diurnal.
Annual aberration is that produced by the earth's motion in its
orbit and is the same for all observers. Diurnal aberration is
due to the earth's daily rotation about its axis, and is different
in different latitudes, because the speed of a point on the earth's
surface is greatest at the equator and diminishes toward the pole.
If v represents the velocity of the earth in its orbit and V the
velocity of light, then when CB is at right angles to AB the
displacement is a maximum and
tan a = >
where a is the angular displacement and is called the "constant
of aberration." Its value is about 20."$. If CB is not per-
pendicular to AB, then
tan a = sin B,
where a is the angular displacement and B is the angle ABC.
Referring to Fig. 2, make a sketch showing the path which Jupiter appears
to describe, in the plane of its motion, but considering the earth as a fixed
point on the diagram.
DEFINITIONS POINTS AND CIRCLES OF REFERENCE
10. The following astronomical terms are in common use and
are necessary in denning the positions of celestial objects on the
sphere by means of spherical coordinates.
A vertical line at any point on the earth's surface is the direc-
tion of gravity at that point, and is shown by the plumb line
or indirectly by means of the spirit level (OZ, Fig. 12).
If the vertical at any point be prolonged upward it will pierce
the sphere at a point called the Zenith (Z, Fig. 12). This point
is of great importance because it is the point on the sphere which
indicates the position of the observer on the earth's surface.
The point where the vertical prolonged downward pierces the
sphere is called the Nadir (N', Fig. 12).
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 are great circles passing through the zenith
and nadir. They all cut the horizon at right angles (HZJ,
POINTS AND CIRCLES OF REFERENCE 1 5
Parallels of altitude, or almucantars, are small circles parallel
to the horizon (DFG, Fig. 12).
If the earth's axis of rotation be produced indefinitely it will
pierce the sphere in two points called the celestial poles (PP'
The celestial equator is a great circle of the celestial sphere
cut by a plane through the centre of the earth perpendicular to
FIG. 12. THE CELESTIAL SPHERE
the axis of rotation (QWRE, Fig. 12). It is everywhere 90
from the poles. A parallel plane through the observer cuts the
sphere in the same circle.
1 6 PRACTICAL ASTRONOMY
Hour Circles are great circles passing through the north and
south celestial poles (PVP f , Fig. 12).
Parallels of Declination.
Small circles parallel to the plane of the equator are called
parallels of decimation (BKC, Fig. 12).
The meridian is the great circle passing through the zenith and
the poles (SZPL, Fig. 12). It is at once an hour circle and a
vertical circle. It is evident that different observers will in
general have different meridians. The meridian cuts the horizon
in the north and south points (N,S, Fig. 12). The intersection
of the plane of the meridian with the horizontal plane through
the observer is the meridian line used in plane surveying.
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).
The ecliptic is the great circle on the celestial sphere which
the sun's centre appears to describe during one year (AMVL,
Fig. 12). Its plane is the plane of the earth's orbit; it is inclined
to the plane of the equator at an angle of about 23 27', called the
obliquity of the ecliptic.
The points of intersection of the ecliptic and the equator are
called the equinoctial points or simply the equinoxes. That
intersection at which the sun appears to cross the equator when
going from the south side to the north side is called the Vernal
Equinox, or sometimes the First Point of Aries (V, Fig. 12).
The sun reaches this point about March 21. The other inter-
section is called the Autumnal Equinox (A, Fig. 12).
The points on the equator midway between the equinoxes are
called the winter and summer solstices.
POINTS AND CIRCLES OF REFERENCE \J
The great circle through the poles and the equinoxes is called
FIG. 12. THE CELESTIAL SPHERE
the equinoctial colure (PVP', Fig. 12). The great circle through
the poles and the solstices is called the solstitial colure.
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
SYSTEMS OF COORDINATES ON THE SPHERE
ii. Spherical Coordinates.
The direction of a point in space may be denned by means
of two spherical coordinates, that is, by two angular distances,
measured on a sphere along arcs of two great circles which
cut each other at right angles. Suppose that it is desired to
locate C (Fig. 13) with reference to the plane OAB and the line
B -^^A. PRIMARY
FIG. 13. SPHERICAL COORDINATES
OA, O being the origin of coordinates. Pass a plane OBC
through C and perpendicular to OAB: these planes will intersect
in the line OB. The two angles which fix the position of C, or
the spherical coordinates, are BOC and AOB. These may be
regarded as the angles at the centre of the sphere or as the arcs
BC and AB. In every system of spherical coordinates the two
coordinates are measured, one on a great circle called the primary,
and the other on one of a system of great circles at right angles
to the primary called secondaries. There are an infinite number
of secondaries, each passing through the two poles of the primary.
The coordinate measured from the primary is. an arc of a
SYSTEMS OF COORDINATES ON THE SPHERE 19
secondary circle; the coordinate measured between the secondary
circles is an arc of the primary.
12. Horizon System.
In this system the primary circle is the horizon and the sec-
ondaries are vertical circles, or circles passing through the zenith
and nadir. The first coordinate of a point is its angular distance
above the horizon, measured on a vertical circle; this is called
the Altitude. The complement of the altitude is called the
Zenith distance. The second coordinate is the angular distance
on the horizon between the meridian and the vertical circle
through the point; this is called the Azimuth. Azimuth may be
reckoned either from the north or the south point and in either
direction, like bearings in surveying, but the custom is to reckon
it from the south point right-handed from o to 360 except for
stars near the pole, in which case it is more convenient to reckon
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
equator, measured on an hour circle; it is called the Declination.
Declinations are considered positive when north of the equator,
negative when south. The complement of the declination is
called the Polar Distance. The second coordinate of the point
is the arc of the equator between the vernal equinox and the foot
of the hour circle through the point; it is called Right Ascension.
Right ascension is measured from the equinox eastward to the
hour circle through the point in question ; it may be measured in
degrees, minutes, and seconds of arc, or in hours, minutes, and
FIG. 15. THE EQUATOR SYSTEM
seconds of time. In Fig. 15 the decimation of the star S is AS;
the right ascension is VA.
Instead of locating a point by means of declination and right
ascension it is sometimes more convenient to use declination
and Hour Angle. The hour angle of a point is the arc of the
SYSTEMS OF COORDINATES ON THE SPHERE
equator between the observer's meridian and the hour circle
through the point. It is measured from the meridian westward
(clockwise) from o h to 24* or from o to 360. In Fig. 16 the
declination of the star S is AS (negative); the hour angle is
FIG. 1 6. HOUR ANGLE AND DECLINATION
MA. It is evident that the hour angles of all points on the
celestial sphere are always increasing.
These three systems are shown in the following table.
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 O the observer. The point Z is the
observer's zenith, so that the latitude on the sphere is the arc
E'Z, which evidently will contain the same number of degrees
as EO. The complement of the latitude is called the Co-latitude.
SYSTEMS OF COORDINATES ON THE SPHERE 23
The terrestrial longitude of the observer is the arc of the equator
between the primary meridian (usually that of Greenwich) and
the meridian of the observer. On the celestial sphere the longi- "1
tude would be the arc of the celestial equator contained between (
two hour circles whose planes are the planes of the two terrestrial^
1 6. Relation between the Two Systems of Coordinates.
In studying the relation between different points and circles
on the sphere it may be convenient to imagine that the celestial
sphere consists of two spherical shells, one within the other.
FIG. 18. THE SPHERE SEEN FROM THE OUTSIDE
The outer one carries upon its surface the ecliptic, equinoxes,
poles, equator, hour circles and all of the stars, the sun, the moon
and the planets. On the inner sphere are the zenith, horizon,
vertical circles, poles, equator, hour circles, and the meridian.
The earth's daily rotation causes the inner sphere to revolve,
24 PRACTICAL ASTRONOMY
while the outer sphere is motionless, or, regarding only the
apparent motion, the outer sphere revolves once per day on its
axis, while the inner sphere appears to be motionless. It is
evident that the coordinates of a fixed star in the first equatorial
system (Declination and Right Ascension) are practically always
the same, whereas the coordinates in the horizon system are
continually changing. It will also be seen that in the first
equatorial system the coordinates are independent of the ob-
server's position, but in the horizon system they are entirely
dependent upon his position. In the second equatorial system
one coordinate is independent of the observer, while the other
(hour angle) is not. In making up catalogues of the positions
of the stars it is necessary to use right ascensions and declina-
tions in defining these positions. When making observations
E R w
FIG. 19. THE SPHERE SEEN FROM THE EARTH (LOOKING SOUTH)
with instruments it is usually simpler to measure coordinates
in the horizon system. Therefore it is necessary to be able to
compute the coordinates of one system from those of another.
The mathematical relations between the spherical coordinates
are discussed in Chapter IV.
SYSTEMS OF COORDINATES ON THE SPHERE 25
Figs. 18, 19, and 20 show three different views of the celestial
sphere with which the student should be familiar. Fig.. 18 is
the sphere as seen from the outside and is the view best adapted
to showing problems in spherical trigonometry. The star 5 has
the altitude RS, azimuth S'R, hour angle Mm, right ascension
Vm, and declination mS; the meridian is ZMS' . Fig. 19 shows
a portion of the sphere as seen by an observer looking southward;
the points are indicated by the same letters as in Fig. 18. Fig. 20
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.
Questions and Problems
1 . What coordinates on the sphere correspond to latitude and longitude on the
2. Make a sketch of the sphere and plot the position of a star having an altitude
of 20 and an azimuth of 250. Locate a star whose hour angle is i6 h and whose
declination is 10. Locate a star whose right ascension is 9** and whose declina-
tion is N. 30.
3. If a star is on the equator and also on the horizon, what is its azimuth? Its
altitude? Its hour angle? Its declination?
RELATION BETWEEN COORDINATES
17. Relation between Altitude of Pole and Latitude of Ob-
In Fig. 21, SZN represents the observer's meridian; let P be
the celestial pole, Z the zenith, E the point of intersection of the
meridian and the equator, and
N and S the north and south
points of the horizon. By the
definitions, OZ (vertical) is
perpendicular to SN (horizon)
and OP (axis) is perpendicular
to EO (equator). Therefore
the arc PN = arc EZ. By the
definitions, EZ is the declination of the -zenith, or the latitude,
and PN is the altitude of the celestial pole. Hence the altitude
of the pole is always equal to the latitude of the observer. The same
relation maybe seen from Fig. 22, in which P is the north pole
of the earth, OH is the plane of the horizon, the observer being
at O, EQ is the earth's equator, and OP' is a line parallel to CP
and consequently points to the celestial pole. It may readily
be shown that ECO, the observer's latitude, equals HOP', the
altitude of the celestial pole. A person at the equator would
see the north celestial pole in the north point of his horizon and
the south celestial pole in the south point of his horizon. If he
travelled northward the north pole would appear to rise, its
altitude being always equal to his latitude, while the south pole
would immediately go below his horizon. When the traveller
reached the north pole of the earth the north celestial pole
would be vertically over his head.
To a person at the equator all stars would appear to move
vertically at the times of rising and setting, and all stars would
be above the horizon i2 h and below i2 h during one revolution
FIG. 23. THE RIGHT SPHERE
of the sphere. All stars in both hemispheres would be above
the horizon at some time every day. This is called the " right
sphere" (Fig. 23).
If a person were at the earth's pole the celestial equator would
coincide with his horizon, and all stars in the northern hemi-
sphere would appear to travel around in circles parallel to the
horizon; they would be visible for 24* a day, and their altitudes
would not change. The stars in the southern hemisphere would
never be visible. The word north would cease to have its usual
RELATION BETWEEN COORDINATES
meaning, and south might mean any horizontal direction. The
longitude of a point on the earth and its azimuth from the
Greenwich meridian would then be the same. This is called
the "parallel sphere" (Fig. 24).
At all points between these two extreme latitudes the equator
cuts the horizon obliquely. A star on the equator will be above
FIG. 24. THE PARALLEL SPHERE
the horizon half the time and below half the time. A star north
of the equator will (to a person in the northern hemisphere) be
above the horizon more than half of the day; a star south of the
equator will be above the horizon less than half of the day. If
the north polar distance of a star is less than the observer's north
latitude, the whole of the star's diurnal circle is above the hori-
zon, and the star will therefore remain above the horizon all
of the time. It is called in this case a circumpolar star (Fig.
25). The south circumpolar stars are those whose south polar
distances are less than the latitude; they are never visible to an
observer in the northern hemisphere. If the observer travels
north until he is beyond the arctic circle, latitude 66 33' north,
then the sun becomes a circumpolar at the time of the summer
solstice. At noon the sun would be at its maximum altitude;
at midnight it would be at its minimum altitude but would still
be above "the horizon. This is called the " midnight sun."
FIG. 25. CIRCUMPOLAR STARS
18. Relation between Latitude of Observer and the Declina-
tion and Altitude of a Star on the Meridian.
The relation between the latitude, altitude, and declination
at the instant when a star is crossing the observer's meridian may
be seen from Fig. 26. Let A be a star on the meridian, south of
the zenith and north of the equator; then
EZ = L, the latitude,
EA = D, the declination,
SA = h, the altitude,
ZA = z, the zenith distance.
From the figure
ZA = EZ - EA
z = L - D
h = 90 - (L - D};
L = 90 - (h - D).
RELATION BETWEEN COORDINATES 3 1
If A is south of the equator the declination is considered
negative, so the same equation will hold true for this case.
FIG. 26. STAR ON THE MERIDIAN
If the star is north of the zenith, as at B, it wil^be more con-
venient to use the polar distance,* = 90 D.
In this case NP = NB - PB
or L h p.
If B is below the pole the equation is
L = h + p.
19. The Astronomical Triangle.
By joining the pole, zenith, and any star 5 on the sphere
by arcs of great circles we obtain a triangle from which the rela-
tion existing among the spherical coordinates may be obtained.
This triangle is so frequently employed in astronomy and navi-
gation that is it called the "astronomical triangle" or the "PZS
triangle." In Fig. 27 the arc PZ is the complement of the
latitude, or co-latitude; arc ZS is the zenith distance or comple-
ment of the altitude; arc PS is the polar distance or complement
of the declination; the angle P is the hour angle of the star if
.S is west of the meridian, or 360 minus the hour angle if
S is east of the meridian; and Z is the azimuth of S, or 360
minus the azimuth, according as S is west or east of the meridian.
The angle at S is called the parallactic angle ; it is little used in
practical astronomy. If any three parts of this triangle are
known the other three may be calculated. The fundamental
formulae of spherical trigonometry are
cos a = cos b cos c + sin b sin c cos A ,
sin a cos B = cos b sin c sin b cos c cos ^4 ,
sin a sin 5 = sin b sin ^4 .
If we put A = P, B = S, C = Z, a = 90 - h, b = 90 - L,
c = 90 D, then these become
(sin h = sin L sin D + cos L cos D cos P, 
cos /* cos S = sin L cos Z) cos L sin Z) cos P, 
cos A sin 5 1 = cos L sin P. 
If A = P, B = Z, C = S, a = 90 - h, b = 90 - D, c = 90 - "L,
cos h cos Z = sin D cos Z, cos D sin L cos P, [n]
cos A sin Z = cos Z) sin P. 
If ^ = Z, B = S, C = P, a = 90 - D, b = 90 - L, c = 90 - h,
sin D = sin L sin h + cos Z, cos h cos Z, 
cos D cos 5 = sin L cos ^ cos Lsin k cos Z, 
cos D sin 5 = cos L sin Z. 
Other forms may be derived by assigning different values to
the parts of the triangle ABC. The formulae given in the
following chapters may in nearly all cases be derived from
equations  to .
The most common cases arising in the practice of surveying
1. Given the declination, latitude, and altitude, to find the
azimuth and the hour angle.
2. Given the declination, latitude, and hour angle, to find the
azimuth and the altitude.
RELATION BETWEEN COORDINATES
In the following formulae
and also let
P = the hour angle,
'Z = the azimuth,*
^h = the altitude,
z = the zenith distance,
D = the declination,
p = the polar distance,
L = the latitude,
FIG. 27. . THE ASTRONOMICAL TRIANGLE
For computing P any of the following formulae may be used,
sin \ P = y/V sin H* + (L - D)} sin * [z-(L=M\ , [l6]
V I T r\
, ? \ cos L cos Z> /
* In the formulae which follow Z is reckoned from the north (interior angle)
unless otherwise designated.
sin * P = t/fcosjsinfr-j
cos L cos Z)
cos * P =
- />) sin (5 -
cos D cos
vers P =
cos 5 sin (s h)
,sin (s L) cos (5
cos (L D) sin h
cos Z cos 'D
For computing the angle Z (measured from the north point) we
i 7 _ //sin* (* + L - D) cosH* + L + Z?)\
cos L sn 2
in Z = 4 /sin (5 - A) sin (5 - L)\
cos L cos
\ Z = J(cs S co S (s-p)\
y V cos L cos h I
tan i Z =
^ ~ sl 5 ~
cos ^ cos (s p)
^ cos (L + h) + sin Z)
,versZ g * = - r- - -
cos L cos h
While any of these formulae may be used to determine the angle
sought, the choice of formula should depend somewhat upon
the precision with which the angle is denned by the function.
If the angle is quite small it is more accurately found through its
sine than through its cosine; for an angle near 90 the reverse
is the case. On account of the rapid variation of the tangent
an angle is always more precisely determined by this function
than by either the sine or the cosine. The versed sine formulae
require the use of both natural and logarithmic functions, but
are sometimes convenient.
* In this case Z is reckoned from the south.
RELATION BETWEEN COORDINATES 35
For computing the altitude and azimuth the following for-
mulae may be used:
cos M tan P ...
tan Z 8 = -T 77 - . * 26
sin (L M)
, cos Z g
tan k = tan (L-lff
where M is an auxiliary angle such that tan M = - ; Z s is
measured from the south point.
The altitude may also be found from the formula
sin h = cos (L D) 2 cos L cos D sin 2 \ P 
or sin h = cos (L D) cos L cos D vers P, 
which may be derived from Equa. .
If the declination, hour angle, and altitude are given, the
azimuth is found from
. ,, . ^cos D
sin Z = sin P -
= sin P cos D sec h. 
For computing the azimuth of a star near the pole when the
hour angle is known the following formula is frequently used:
tan Z =
cos L tan D sin L cos P
This equation may be derived by dividing  by [n] and then
simplifying the result by dividing by cos D.
Given the latitude and declination, find the hour angle and
azimuth of a star on the horizon. Putting h = o in Equa. 
and  the results are
cos P = tan D tan L 
~ sin D T -,
and cos Z = -- - L33
* For the derivation of this formula see Chauvenet's Spherical and Practical
Astronomy, Vol. I, Art. 14.
A special case of the PZS triangle occurs when a star near the
pole (circumpolar) is at its greatest east or west position, known
as its greatest elongation. At this time the star's bearing or
azimuth is a maximum and its diurnal circle is tangent to the
E HORIZON A N
FIG. 28. STAR AT GREATEST ELONGATON (EAST)
vertical circle through the star (Fig. 28) ; the triangle is conse-
quently right-angled at S.
The formulae for this case are
cos p =
and \^ (sinZ = sin p sec L. j> 
20. Relation between Right Ascension and Hour Angle.
In order to understand the relation between the right ascen-
sion and the hour angle of a point, we may think of the equator
on the outer sphere as graduated into hours, minutes, and seconds
of right ascension, zero being at the equinox and the numbers
increasing toward the east. The equator on the inner sphere is
graduated for hour angles, the zero being at the observer's
meridian and the numbers increasing toward the west. (See
Fig. 29.) As the outer sphere turns, the hour marks on the right
ascension scale will pass the meridian in the order of the numbers.
The number opposite the meridian at any instant shows how far
FIG. 29. RIGHT ASCENSION AND HOUR ANGLE
38 PRACTICAL ASTRONOMY
the sphere has turned since the equinox was on the meridian.
If we read the hour angle scale opposite the equinox, we obtain
exactly the same number of hours. This number of hours (or
angle) may be considered as either the right ascension of the
meridian or the hour angle of the equinox. In Fig. 30 the star
S has an hour angle equal to AB and a right ascension CB. The
sum of these two angles is AC, or the hour angle of the equinox.
The same relation will be found to hold true for all positions of
5. The general relation existing between these coordinates is,
Hour angle of Equinox = Hour angle of Star + Right Ascen-
sion of Star. 
Questions and Problems
1. What is the greatest declination a star may have and culminate south of the
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
5. What are the greatest and least angles made by the ecliptic with the horizon
6. In what latitudes is Vega (Decl. = 38 42' N.) a circumpolar star?
7. Make a sketch of the celestial sphere as it appears to an observer in latitude
20 South at the instant the vernal equinox is on the eastern horizon.
8. Derive formula .
MEASUREMENT OF TIME
21. The Earth's Rotation.
The measurement of intervals of time is made to depend upon
the period of the earth's rotation on its axis. Although it is
probable that this period is not absolutely invariable, yet the
variations are too small to be measured, and the rotation is
assumed to be uniform. The most natural unit of time for
ordinary purposes is the solar day, or the time of one rotation
of the earth with respect to the sun's direction. On account of
the earth's annual motion around the sun the direction of the
reference line is continually changing, and the length of the
solar day is not the true time of one rotation of the earth on its )
axis. For this reason it is necessary in astronomical work to
make use of another kind of time, based upon the actual period of
rotation, called sidereal time (star time).
22. Transit or Culmination.
Every point on the celestial sphere crosses the meridian of an
observer twice during one revolution of the sphere. The instant
when any point on the celestial sphere is on the meridian of an
observer is called the transit, or culmination, of that point over
that meridian. When it is on that half of the meridian contain-
ing the zenith, it is called the upper transit; when it is on the
other half it is called the lower transit. Except in the case of
stars near the elevated pole the upper transit is the only one
visible to the observer; hence when the transit of a star is men-
tioned the upper transit will be understood unless the contrary
23. Sidereal Day.
The sidereal day is the interval of time between two successive
upper transits of the vernal equinox over the same meridian.
40 PRACTICAL ASTRONOMY
If the equinox were absolutely fixed in position, the sidereal day
as thus denned would be the true period of the earth's rotation;
but since the equinox has a slow westward motion caused by the
precessional movement of the axis (see Art. 8), the actual
interval between two transits of the equinox differs about
o s .oi from the true time of one rotation. The sidereal day actu-
ally used in practice, however, is the one denned above and not
the true rotation period. Sidereal days are not used for reckon-
ing long periods of time, dates always being in solar days, so this
error never becomes appreciable. The sidereal day is divided
into 24 hours and each hour is subdivided into minutes and
seconds. When the equinox is at upper transit it is O A , or the
beginning of the sidereal day (sidereal " noon ").
24. Sidereal Time.
The sidereal time at a given meridian at any instant is the
hour angle of the vernal equinox. It is therefore a measure of
the angle through which the earth has turned since the equinox
was on the meridian, and shows the position of the sphere at
the given instant with respect to the observer's meridian.
25. Solar Day.
A solar day is the interval of time between two successive upper
transits of the sun's centre over the same meridian. It is divided
into 24 hours, each hour being divided into minutes and seconds.
When the sun is on the upper side of the meridian (upper
transit) it is noon, or o h solar time. When it is on the lower side
of the meridian it is midnight.
26. Solar Time.
The solar time at a given meridian at any instant is the hour
angle of the sun's centre at that instant. This hour angle is a
measure of the angle through which the earth has turned with
respect to the sun's direction, and consequently is a measure of
the time elapsed since the sun was on the meridian.
Since the earth revolves around the sun in an elliptical orbit
in accordance with the law of gravitation, the apparent angular
motion of the sun is not uniform, and the days are therefore of
MEASUREMENT OF TIME 41
unequal length at different seasons. In former times, when sun
dials were considered sufficiently accurate for measuring time,
this lack of uniformity was not important. Under modern
conditions, which demand accurate measurement of time by the
use of clocks, an invariable unit of time is essential. As a con-
sequence, the time adopted for common use is that kept by a
fictitious sun, or mean sun, which is conceived to move at a
uniform rate along the equator,* its speed being such that it
makes one apparent revolution around the earth in the same time
as the true sun (i.e., one year). The fictitious sun is so placed
that on the whole it precedes the true sun as much as it follows
it. The time indicated by the position of the mean sun is called
mean solar time, or simply mean time. The time indicated by
the position of the real sun is called apparent solar time and is
the time shown by a sun dial.
27. Equation of Time.
Since observations made on the sun for the purpose of deter-
mining the time can give apparent time only, it is necessary to be
able to find at any instant the exact relation between apparent
and mean time. The difference between the two, which varies
from +i6 m to i6 m (nearly), is called the equation of time.
This quantity may be found in the Nautical Almanac for each
day of the year.
This difference between the two kinds of time is due to several
causes, the chief of which are (i) the inequality of the earth's
angular motion in the orbit, and (2) the fact that the true sun
is on the ecliptic while the mean sun is on the equator. In the
winter, when the earth is nearest the sun, the rate of angular
motion about the sun must be greater than in summer in order
that the radius vector shall describe equal areas in equal inter-
vals of time. (See Fig. 6 and Art. 6.) The sun will then appear
* This statement is true in a general way, but the motion is not strictly uniform
because the motion of the equinox itself is variable. The angle from the equinox
to the " mean sun " at any instant is the sun's " mean longitude " (along the
ecliptic) plus periodic terms.
42 PRACTICAL ASTRONOMY
to move eastward in the sky at a faster rate than in summer,
and its daily revolution about the earth will be slower. This
delays the instant of apparent noon, making the apparent solar
days longer than their average, and therefore a sun dial will
" lose time." About April i the sun is moving at its average
speed and the sun dial ceases to lose time; from this date until
about July i the sun dial gains on mean time, making up what
it lost between Jan. i and April i. During the other half of the
year the process is reversed; the sun dial gains from July i to
Oct. i and loses from Oct. i to Jan. i. The maximum difference
in time due to this cause is about 8 minutes, either + or .
The second cause of the equation of time is illustrated by
Fig. 31. Assume that point S' (sometimes called the " first
mean sun") moves uniformly along the ecliptic at the average
rate of the true sun; the time as indicated by this point will
evidently not be affected by the eccentricity of the orbit. If
the mean sun S (also called " the second mean- sun") starts at
V, the equinox, at the same instant that S' starts, then the arcs
VS and VS' are equal, since both points are moving with the
same speed. By drawing hour circles through these two points
it will be seen that these hour circles do not coincide except
when the points are at the equinoxes or at the solstices. Since
the points are not on the same hour circle they will not cross the
meridian at the same time, the difference in time being repre-
MEASUREMENT OF TIME
sented by the arc aS. The maximum length of aS is about
10 minutes of time, which may be either + or . The com-
bined effect of these two causes, or the equation of time, is shown
in the following table.
TABLE A. EQUATION OF TIME FOR 1910.
+ yn 26s
+ ym 2 ?s
+ II* 02
+ I3 TO 22*
+ 13 4-i
+ 14 24
+ IT. S.Q
+ 12 ^8
+ 10 36
+ 7 48
+ 4 4^
+ 4 08
+ 1 31
- o <S
- 2 48
- o ?7
+ i 08
+ * i;
+ 3 27
-'- ^ 01
+ 6 06
+ 6 16
+ 6 n
+ 5 IQ
+ * 26
+ o 46
+ o oq
- 2 48
+ 2 21
28. Conversion of Apparent Time into Mean Time and vice
Apparent time may be converted into mean time by adding
or subtracting the equation of time at the instant. Since the
equation of time is given in the Nautical Almanac for Greenwich
noon its value at the desired instant must be found by adding
\ or subtracting the increase or decrease since Greenwich noon.
Example i. Find the mean time of the sun's transit over the meridian of Boston
on June 30, 1910. The apparent time at Boston is 12^ oo m oo s M. at the instant
of the transit of the sun's centre, and this corresponds to 4 h 44 iS s apparent
time at Greenwich, since the longitude of Boston is 4^ 44 i8 s west of Greenwich.
The equation of time at Greenwich Apparent Noon is 3"* i4 s -92 (to be added to
apparent time); the hourly change is o s .5oo (increasing). The correction to be
applied to the equation of time is 4^.74 X o s .5oo = 2 s -37, making the equation of
time at Boston noon 3"* 17*. 29.
L. A. T.* = 12^00 oo s .oo
Equa. of. T =
3 17 -29
L. M. T. = 12* 03 1 7 s . 29
* A list of abbreviations will be found on p. 191.
44 PRACTICAL ASTRONOMY
Example 2. Find the local apparent time at Boston at 2 P.M. (local mean time)
Oct. 28, 1910. The Greenwich Mean Time corresponding to 2 P.M. local mean
time is 6 h 44"* i8 s P.M. The equation of time at G. M. N. Oct. 28, 1910, is
i6 m 04 s .29 (to be added to mean time); the hourly increase is o s .2o8. The correc-
tion to the equation of time is 6^.7 4 X o*.2o8 = i s .4o. The equation of time at
2 P.M. is therefore i6 TO c>5 s .69.
L. M. T. = 2 h oo m oo s .oo
Equa. of T. = 16 05 .69
L. A. T. = 2 ft i6 m o5 s .69
29. Astronomical and Civil Time.
For ordinary purposes it is found convenient to divide the
solar day into two parts of i2 h each; from midnight to noon is
called A.M. (ante meridiem), and from noon to midnight is
called P.M. (post meridiem). The date changes at the instant
of midnight. This mode of reckoning time is called Civil Reck-
oning. In astronomical work this subdivision of the day is not
convenient. For simplicity in calculation the day is divided
into 24^, numbered consecutively from o h to 24^. As it is not
convenient to have the date change during the night, the astro-
nomical date begins at noon or o h . This is called Astronomical
Time. In using the Nautical Almanac it should be remembered ^
that it is necessary to change the date and hours to astronomical y
time before taking out the desired data. In order to change *
from one kind of time to the other it is only necessary to remem-
ber that the astronomical slay begins at noon of the civil day of
the same date; that is, in the afternoon the dates and the hours
will be the same, but in the forenoon the astronomical date is
one day less and the hours are 1 2 greater than in the civil time.
Astr. Time May 10, 15* = Civil Time May u, 3* A.M.
" Jan. 3, 7* = " Jan. 3, 7 h P.M.
From these examples the following rules may be derived :
To change Civil Time to Astronomical Time,
If A.M., add i2 h and drop i day from date, and drop the A.M.
If P.M., drop the P.M.
MEASUREMENT OF TIME 45
To change Astronomical Time to Civil Time*
If less than i2 h , mark it P.M.
If greater than i2 h , subtract i2 h , add i day to date, and mark
30. Relation between Longitude and Time.
The hour angle of the sun at any given meridian at a given
instant is the local solar time at that meridian, and will be
apparent or mean time according as the true sun or the mean
sun is considered. The hour angle of the sun at Greenwich at
the same instant is the corresponding Greenwich solar time.
The difference between the two hour angles is the longitude
of the place from Greenwich, expressed either in degrees or in
hours according as the hour angles themselves are expressed
in degrees or in hours. Similarly the difference in local solar
time of any two places at a given instant is their difference in
longitude in hours, minutes, and seconds. In Fig. 32, AC is
the hour angle of the sun at Greenwich (G), or the Greenwich
solar time. EC is the hour angle of the sun at the meridian
through P, or the local solar time at P. The difference, AB,
is the longitude of P west of Greenwich. It should be observed
that the reasoning is exactly the same whether C represents the
true sun or the fictitious sun. The same result would also be
found if the point C were to represent the vernal equinox. The
arc AC would then be the hour angle of the equinox, i.e., the
Greenwich Sidereal Time. BC would be the Local Sidereal
Time, and AB the difference in longitude. The measurement
of longitude is therefore independent of the kind of time used,
because in each case the angular distances to A and B are meas-
ured from the same point C on the equator, and the difference in
these angles does not depend upon the position of this point
nor upon the speed with which this point has moved up to the
position at C.
* The student may find it helpful to plot the time along a straight line, and to
write two sets of numbers, one for Civil Dates and the other for Astronomical Dates.
The difference in the sidereal times at meridian A and meridian
B (Fig. 32) is the interval of sidereal time during which a star
would go from A to B. Since the star requires 24 sidereal hours
to travel from meridian A to meridian A again, the time interval
from A to B bears the same relation to 24* that the longitude
difference bears to 360. The difference in the mean solar times
at A and B is the number of mean solar hours that the^sun
would take to go from A to B, and since the sun takes 24 solar
hours to go from A to A again, the time interval from A to B
bears the same ratio to 24 solar hours as when sidereal time was
used. The difference in longitude is therefore correctly given
when either sidereal or solar times are compared.
The method of changing from Greenwich to local time and the
reverse is illustrated by the following examples.
Example i. The Greenwich astronomical time is 7*40"* io s .o. Required the
local time at a meridian 4 h $o m 2i s .o West.
G. M. T. = 7 h 40 io s .o
Long. West = 4 5 21 -Q
L. M. T. = 2 h 49 49 s .o (P.M.)
MEASUREMENT OF TIME 47
Example 2. The Greenwich mean time is 3^ 20 i6 s .5. Required the local
mean time at a place whose longitude is 120 10' West.
G. M. T. + 24 h = 2j h 20 i6 s .5
Long. West = 8 h oo m 40 s .o
L. M. T. = 19* IQ W 36 s .s
= 7 A i9 TO 36 s .5A.M.
Example 3. The mean time at a place 3^ East longitude is io h A.M. Required
the Greenwich mean time.
L. M. T. = 22 h oo m oo s . o
Long. East = 3^ oo m oo s .o
G. M. T. = ig h oo m oo s . o
= J h 00 00 S . O A.M.
Since a circle may be divided either into 24^ or into 360, the
relation between these two units is constant. From the fact that
2 4 h = 360
we have also i h = 15,
i w = 15',
i s - IS"-
The following equivalents are also convenient:
By means of these two sets of equivalents time may be con-
verted into degrees, or the contrary, without writing down the
intermediate steps. In the following examples the intermediate
steps are written down in order to show the process followed.
Example i. Convert 6 h 35 51* into degrees.
6 h = 90
35 W = 32 m + 3 m = 8 45'
5 i s = 4 8 s + 3 s = __ .12' 45"
Total = 98 57' 45"
Example 2. Convert 47 17' 35" into hours.
47 = 45 -(- 2 = 3 A o8 m
17' = 15' + 2 ' = oi m o8 8
35" = 30" + 5" = 02 .33
Total = 3" 09 w i o s . 33
It should be observed that the relation 15 = i h is quite
independent of the length of time that has elapsed. A star
takes one sidereal hour to move over 15 of hour angle; the sun
takes one solar hour to move over 15 of hour angle. In the
sense in which it is used here, i h means an angle, and not an
absolute interval of time.
31. Relation between Sidereal Time, Right Ascension, and
Hour Angle of any Point at a Given Instant.
In Fig. 33 the hour angle of the equinox, or local sidereal time
at the meridian through P, is the arc A V. The hour angle of
the star 5 at the meridian through P is the arc AB. The right
ascension of the star 5 is the arc VB. It is evident from the
AV =VB + AB,
or (S.R.+ Pj 
where R = the right ascension and P = the hour angle of the
point S, and 5 = the sidereal time; or, in words,
Sidereal Time = Right Ascension -\-Hour Angle. 
MEASUREMENT OF TIME 49
This relation is a perfectly general one and will be found to hold
true for all points on the sphere, provided it is agreed to reckon
the sidereal time beyond 24* when necessary. For example, if
the hour angle is io h and the right ascension is 2o h , the resulting
sidereal time is 30*. This means that the equinox has made a
complete revolution and has gone 6 h , or 90, on the next revolu-
tion; the actual reading of the sidereal clock would be 6 h . In
the reverse case, when it is necessary to subtract 2O A from 6*
to obtain the hour angle, the 6 h must first be increased by 24*
and the right ascension subtracted from the sum to obtain the
hour angle, io h .
32. Star on the Meridian.
At the instant when the star is on the meridian its hour angle
is O A and the equation becomes
Sidereal Time = Right Ascension; 
that is, the right ascension of a star equals the local sidereal time
at which that star crosses the meridian. (See Art. 20, p. 36.)
33. Relation between Mean Solar and Sidereal Intervals of
It has already been stated that on account of the earth's
orbital motion the sun has an apparent eastward motion among
the stars of nearly i per day. This eastward movement of the
sun makes the intervals between the sun's transits greater by
nearly 4"* than the intervals between the transits of the equinox,
that is, the solar day is nearly 4 longer than the sidereal day.
In Fig. 34 let C and C' be the positions of the earth on two
consecutive days. When the observer is at O it is local noon.
After the earth makes one complete rotation, the observer will
be at O', and the sidereal time will be exactly the same as it was
the day before when he was at O. But the sun's direction is
now C'O", so the earth must turn through the angle O'C'O"
in order to bring the sun again on the observer's meridian.
Since this angle is about i it takes about 4 longer to complete
the solar day than it does to complete the sidereal day. Since
5 PRACTICAL ASTRONOMY
each kind of day is subdivided into hours, minutes, and seconds,
all of these units in solar time will be proportionally longer than
the corresponding units of sidereal time. If two clocks, one
regulated to mean solar time and the other to sidereal time, were
started at the same instant, both reading o h , the sidereal clock
would immediately begin to gain on the solar clock, the gain
being exactly proportional to the time interval, that is, about io s
per hour, or more nearly 3"* 56* per day.
In order to find the exact relation between the two kinds of
time it should be observed that the number of sidereal days in
the year is exactly one greater than the number of solar days,
because the sun comes back to the equinox at the end of one
year. The length of the tropical* year is found to be 365.2422
* The tropical year is the interval of time between two successive passages of
the sun over the vernal equinox. The sidereal year is the interval between two
passages of the sun across the hour circle through a fixed star on the equator. On
account of the movement of the equinox caused by precession, the tropical year is
about 2O TO shorter than the sidereal year.
MEASUREMENT OF TIME 51
mean solar days. The relation between the two kinds of day is
366.2422 sidereal days = 365.2422 solar days, 
or i sidereal day = 0.99726957 solar day, 
and i solar day = 1.00273791 sidereal days. 
Equations  and  may be written
24 h sidereal time = (24* 3 55 S .9O9) mean solar time,
24* mean solar time = (24* + 3 m 56^.555) sidereal time.
These equations may be put in more convenient form for com-
putation by expressing the difference in time as a correction
to be applied to any interval of time to change it from one kind
of unit to the other. If Im is a mean solar interval and 7 S the
corresponding number of sidereal units, then
I s = I m + .00273791 X I m 
and Im = I s - .00273043 X / s . 
Tables II and III are constructed by multiplying different values
of I m and I 8 by these constants. More extended tables may be
found in the Nautical Almanac. The use of Tables II and III
is illustrated by the following examples.
Reduce 9^ 23'" 5i s .oof sidereal time to the equivalent number
of solar units. From Table II, opposite 9^ is the correction
i m 28 S .466; opposite 23 in the 4th column is 3". 768; and
opposite 5i s in the last column is o s .i39. The sum of these
three partial corrections is i m 32 S .373, which is the amount to
be subtracted from g h 23"* 5i s .o to reduce it to the equivalent
solar interval, g h 22 1 8 s . 627.
Reduce j h io m solar time to sidereal time. The correction for
7 , Table III, is + i m o8 s .995, and for io m is i s .643. The sum,
i m 1 0^.638, added to 7* io m gives 7* u w io s .638 of sidereal time.
This reduction may be made approximately by the following
rule: the correction equals io 8 per hour diminished by i 8 for
every 6* in the interval. The correction for 6 h would be
6 X io 8 i s = 59 s . This rule is based on a change of 3"*
56* per day. For changing solar into sidereal the error is
o s .o23 per hour; for sidereal into solar the error is o s .oo4 per
It should be kept in mind that the .conversion of time discussed
in this article concerns the change from one kind of unit to
another, like changing from yards to metres, and is not the same
as changing from the local sidereal time to the local solar time
at a particular instant.
34. Relation between Sidereal Time and Mean Solar Time
at any Instant.
If in Fig. 33, Art. 31, the point B is taken to represent the
mean sun, then equation  becomes
S = R s + P s , 
where R s and P s are the right ascension and the hour angle of
the mean sun at the instant considered. P s is the local mean
time by the definition given in Art. 26. If the equation is
S - P s = R a , 
then, since the value of the right ascension R s does not depen
upon the time at any particular meridian, but only upon the
absolute instant of time considered, it is evident that the differ-
ence between sidereal time and mean time at any instant is the
same for all places on the earth. The actual values of S and
P a will of course be different at different meridians, but the
difference between the two is a constant for all places for the
given instant. In order that Equa.  shall hold true it is
essential that R s and P s shall refer to the same position of the
sun, that is, to the same absolute instant of time. The right
ascension of the sun obtained from the Nautical Almanac is its
value at the instant of the Greenwich Mean Noon preceding,
that is, at the beginning of the astronomical day at Greenwich.*
* The dates are always in mean solar days, not in sidereal days.
MEASUREMENT OF TIME
To reduce this right ascension to its value at the desired instant
it is necessary to multiply the hourly increase in the right ascen-
sion of the mean sun by the number of solar hours elapsed since
the instant of Greenwich Mean Noon. The hourly increase in
the right ascension of the mean sun is constant and is evidently
equal to the correction in Table III, for the difference between
sidereal and solar time is caused by the sun's motion, and the
amount of the difference for any number of hours is exactly
equal to the increase in the right ascension. If it is desired to
find the increase for any number of solar hours, Table III should
be used; for sidereal hours use Table II. Equation  may
5 = R s + P s + C, 
where R s refers to the instant of the preceding local mean noon,
and C is the correction (Table III) to reduce P s to a sidereal
interval, or to reduce R s to its value at the time P s .
In Fig. 35 suppose that the sun S and a star 5" passed the
meridian M at the same instant, and at the mean time P s it is
desired to compute the sidereal time. Since the sun is moving
at a slower rate than the star, it will describe the arc MS ( = P a )
54 PRACTICAL ASTRONOMY
while the star moves from M to S'. The arc SS', or C, repre-
sents the gain of sidereal on mean time in the mean time interval
MS or P s . But S' is the position of the sun at noon, so that
VS f is the sun's right ascension at the preceding mean noon, or
R s . The right ascension desired is VS, so R s must evidently
be increased by the arc SS', or C.
If it is desired to find the mean solar time corresponding to
a given instant of local sidereal time,, the equation is
Sidereal internal from noon = S R s , 
or Mean time = P s = S R s C', 
where C' is the correction from Table II to reduce 5" Rs to a
solar interval, and represents the increase in the sun's right
ascension in 5 R s sidereal hours.
To find the Greenwich Sidereal Time corresponding to Greenwich Mean Time
-9* 22 m i8 s .6o on Jan. 7, 1907. The right ascension of the mean sun at Greenwich
Mean Noon is found from the Nautical Almanac to be 19^ 03"* 36 s -38. The cor-
rection to reduce 9* 22 i8 8 .6o to sidereal time (Table III) is +i m 32 8 -37. Then,
applying Equa. ,
R s = i9 A 03"36 8 .38
P s = 9 22 18 .60
C = _ i 32 -37
S = 2S h 2 7 m 27 S . 3S
Sidereal Time = 4* 27"* 27^.35
To find the Greenwich Mean Solar Time when the Greenwich Sidereal Time is
4 h 27 27*.3S on Jan. 7, 1907.
S = 2& h 2 7 m 2 7 S .35
R s = 19 03 36 .38
5 - R 9 = 9 23 50 .97
C' (Table H) = -i 32.37
Mean Time = 9* 22 TO i8*.6o
If the change from sidereal to solar time (or vice versa) is to be
made at any meridian other than Greenwich, the right ascension
of the sun for local noon must be found by multiplying the
increase per solar hour by the number of solar hours since Green-
wich noon, that is, by the number of hours in the longitude, and
MEASUREMENT OF TIME 55
adding this to the value of R s from the Almanac if the place is
west of Greenwich, subtracting if east.* The correction may
be taken from Table III. If the sidereal time in the above
example is assumed to be the time at a meridian 5 A (75) west of
Greenwich, the computation would be modified as follows:
R, f = ig h 03 3 6 8 . 3 8
Correction for 5^ longitude = 49 .28
R s = 19 04 25 .66
S = 28 27 27 .35
5 R s = 9 23 01 .69
C - i 32 .24
Local Mean Time = g h 2i m 29^45
It is evident that at the instant of mean noon P s = o and
R s = S. At mean noon, therefore, the sidereal time equals
the right ascension of the mean sun. This quantity will be
found in the Almanac under both headings, " Sidereal Time of
Mean Noon" and " Right Ascension of the Mean Sun." (See
The reduction of mean solar time to sidereal time, or the re-
verse, may be made also by first changing the given local time
to the corresponding instant of Greenwich time, then making
the transformation as before, and finally changing back to the
meridian of the place. Take, for example, the case given on
Local Sidereal Time = 28^ 27 m 27 s .35
Longitude = 5 oo oo
Greenwich Sidereal Time = 33 27 27 .35
R s at Gr. M. Noon = 19 03 36 .38
Sidereal Interval from Noon = 14 23 50 .97
C' = 2 21 .52
Greenwich Mean Time = 14 21 29 .45
Longitude = 5 oo oo
Local Mean Time = g h 2i OT 29 s .45
The result agrees with that obtained by the former method.
This method is quite as simple as the preceding, especially when
* It should be remembered that the sun's R. A. is always increasing.
56 PRACTICAL ASTRONOMY
Standard Time is to be computed, for the final correction will
always be a whole number of hours. Care should be taken
always to use the right ascension of the sun at the noon preced-
ing the given time. Suppose that the instant of io h A.M. May 5
is to be converted into sidereal time, the longitude of the place
being 4 h 4.4 i8 s west. Civil time io h A.M. May 5 = Astr.
time 22 h May 4. If the first method is followed, the right
ascension of the sun employed should be that of noon May 4.
If the reduction is made by first changing to Greenwich time,
then 22* + 4 h 44 m i8 s = 26* 44 i8 s May 4 = 2 h 44 i8 s
May 5. The right ascension for the latter case would be that
for noon of May 5.
35. Standard Time.
From the definition of mean solar time it will be seen that at
any given instant the solar times at two places will differ from
each other by an amount depending upon the difference in the
longitudes. All places will have different local times except
where they happen to be on the same meridian. Previous to the
year 1883 it was customary in this country for each large city
or town to use the mean time at its own meridian, and for all
other places in the vicinity to adopt the same time. Before
railroad travel became extensive this change of time from one
point to another caused no great difficulty, but with the in-
creased amount of railroad and telegraph business these frequent
and irregular changes in time became so inconvenient that in
1883 a uniform system of time was adopted in the United States.
The country is divided into time belts each theoretically 15 in
width; these are known as the Eastern, Central, Mountain and
Pacific time belts, and places in these belts use the mean local
time of the 75, 90, 105 and 120 meridians respectively. The
time at the 60 meridian is called Atlantic time and is used in
the Eastern Provinces of Canada. The actual positions of the
dividing lines between these belts depend upon the positions of
the principal cities and the railroads (see Fig. 36), but the change
of time from one belt to another is always exactly one hour. The
MEASUREMENT OF TIME
58 PRACTICAL ASTRONOMY
minutes and seconds of all clocks are the same as the minutes
and seconds of the Greenwich clock. When it is noon at Green-
wich it is 8 A.M. Atlantic time, 7 A.M. Eastern time, 6 A.M.
Central time, 5 A.M. Mountain time, and 4 A.M. Pacific time.
The change from local to standard time, or the contrary, con-
sists in expressing the difference in longitude between the local
meridian and the standard meridian in units of time, and adding
or subtracting this correction, remembering that the farther
west a place is, the earlier it is in the day at any given instant
Find the standard time at a place 71 west of Greenwich when
the local time is 4 h 20 oo s P.M. In longitude 71 the standard
time would be that of the 75 meridian. The difference in
longitude is 4 = i6 m . Since the standard meridian is west of
the 71 meridian, the time is i6 m earlier than the local time. The
standard time is therefore 4* 04"* oo 8 P.M.
Find the local time at a place 91 west of Greenwich when the
Central time is 9* OO TO oo s A.M. The difference in longitude is
i = 4 m . Since the place is west of the standard meridian, the
time is earlier. The local time is therefore S h 56 oo s A.M.
Standard time is used not only in the United States but in a
majority of the countries of the world; in nearly all cases these
systems of standard time are based on the meridian of Green-
wich as the prime meridian. Germany, for example, uses the
local mean time at the meridian i h east of Greenwich; Japan
uses that of the meridian 9* east of Greenwich ; Turkey, 2 h east
cf Greenwich, etc.
36. The Date Line.
If a person were to start at Greenwich at the instant of noon
and travel westward rapidly enough to keep the sun always on
his meridian he would get back to Greenwich 24* later, but his
own (local) time would not have changed but would have
remained noon all the time. In travelling westward at a slower
rate the same thing occurs, only in a longer interval of time.
MEASUREMENT OF TIME 59
The traveller has to set his watch back every day in order to
keep it regulated to the meridian at which his noon occurs. As
a consequence, his watch has recorded one day less than it has
actually run, and his calendar is one day behind that of a person
who remains at Greenwich. If the traveller goes east he has to
set his watch ahead every day, and after circumnavigating the
globe his calendar is one day ahead of what it should be. In
order that the calendar may be everywhere uniform, it is agreed
to change the date at the meridian 180 from Greenwich. When-
ever a ship crosses the 180 meridian going westward, a day is
omitted from the calendar, and when going eastward a day is
repeated. In practice the change is made at midnight near the
180 meridian, not at the instant of crossing. The date line
actually used does not follow the 180 meridian in all places, but
is deflected so as not to separate the Aleutian islands, and in
the South Pacific ocean it passes east of several groups of islands
so as not to change the date formerly used in these islands.
37. The Calendar.
Previous to the time of Julius Caesar the calendar was based
upon the lunar month, and, as this resulted in a continual change
in the date at which the seasons occurred, the calendar was fre-
quently changed in an arbitrary manner in order to keep the
seasons in their places, the result being extreme confusion in the
dates. In the year 45 B.C. Julius Caesar reformed the calendar
and introduced one based upon a year of 365! days, since called
the Julian calendar. The \ day was taken care of by making
the year contain 365 days, except every 4th year, called leap
year, which contained 366; the extra day was added to February
in such years as were divisible by 4. The year was begun on
Jan. i ; previously it had begun in March. Since the year con-
tains actually 365^ 5^ 48 m 46*, this difference of n m 14* caused
a gradual change in the dates at which the seasons occurred.
After many centuries the difference had accumulated to about
10 days, so in 1582 Pope Gregory XIII ordered that the calendar
should be corrected by dropping ten days and that future dates
should be computed by omitting the 366th day in those leap years
which occurred in century years not divisible by 400; that is,
such years as 1700, 1800 and 1900 should not be counted as leap
years. This is the calendar used at the present time.
Questions and Problems
1. (a) Prove by direct computation of sidereal time from Fig. 37 that
R + P = 24 h + S,
in which R and P are the right ascension and hour angle of the star S, and S is the
sidereal time, or hour angle of V.
(b) Prove the same relation when V is at the point V. (See Art. 31,
2. Prove that the difference in longitude of two points is independent of the
kind of time used, by selecting two points at which the solar time differs by say
3 A , and then converting the solar time at each place into sidereal time.
3. Make a design for a horizontal sun dial for a place whose latitude is 42 21' N.
The gnomon ad (Fig. 38), or line which casts the shadow on the horizontal plane,
must be parallel to the earth's rotation axis; the angle which the gnomon makes
with the horizontal plane therefore equals the latitude. The shadow lines for the
hours (X, XI, XII, I, II, etc.) are found by passing planes through the gnomon
and finding where they cut the horizontal plane of the dial. The vertical plane
adb coincides with the meridian and therefore is the noon (XII*) line. The other
planes make, with the vertical plane, angles equal to some multiple of 15. In
finding the trace dc of one of these planes on the dial it should be observed that the
foot of the gnomon, d , is a point common to all such traces. In order to find another
point c on any trace, or shadow line, pass a plane abc through some point a on the
gnomon and perpendicular to it. This plane (the plane of the equator) will cut
an east and west line cc on the dial. If a line be drawn in this plane making an
MEASUREMENT OF TIME 6 1
angle of n X 15 with the meridian plane, it will cut ce at a point c which is on the
shadow line. Joining c with the foot of the gnomon gives the required line.
In making a design for a sun dial it must be remembered that the west edge of
the gnomon casts the shadow in the forenoon and the east edge in the afternoon;
there will be of course two noon lines, and the two halves of the diagram will be
symmetrical and separated from each other by the thickness of the gnomon. The
dial may be placed in position by levelling the horizontal surface and then com-
puting the watch time of apparent noon and turning the dial so that the shadow is
on the XII A line at the calculated time.
Prove that the horizontal angle bdc is given by the relation
tan bdc = tan P sin L,
in which P is the sun's hour angle and L is the latitude.
4. Why are the sun's and moon's right ascension always increasing?
5. The local apparent time at a point A is io h 30"* A.M. If the equation of
time is + 3"* 25^.8, what is the local mean time? What is the astronomical mean
time at the given instant? Assuming the longitude of A to be 95 West, what is
the Greenwich Mean Time? What is the Central Standard Time? What is the
local mean time at the same instant at a point B in longitude 110 W.? If the
right ascension of the mean sun at G. M. N. is i8 A 4i TO oi s .6, what is the local
sidereal time? What is the Greenwich Sidereal Time?
THE AMERICAN EPHEMERIS AND NAUTICAL
ALMANAC STAR CATALOGUES INTERPOLATION
38. The Ephemeris.
In the problems previously discussed it has been assumed that
the coordinates of celestial objects and various other data men-
tioned are known to the computer. These data consist of
results calculated from observations made with large instruments
at the principal observatories; these results are published by the
government several years in advance in the American Ephemeris
and Nautical Almanac.* The Almanac contains the declinations
and the right ascensions of the sun, moon, planets and stars, as
well as the angular semidiameters, horizontal parallaxes, the
equation of time, and other data required in astronomical cal-
culations. Since all of these quantities vary with the time,
their values are usually given for equidistant intervals of Green-
wich time or of Washington time.
The Almanac is divided into three parts. Part I is computed
for the meridian of Greenwich, and is arranged especially for
the convenience of navigators. Part II is computed for the
meridian of Washington, and is arranged chiefly for the con-
venience of astronomers. Part III contains the data for pre-
dicting phenomena, such as eclipses, occultations, etc. At the
end of the book are certain tables computed especially for the
use of the navigator and the surveyor.
The first page of Part I is headed "At Greenwich Apparent
* Similar publications by other governments are: the Nautical Almanac
(Great Britain), the Berliner Astronomisches Jahrbuch (Germany), the Con-
naissance des Temps (France), and the Almanaque Nautico (Spain).
The word " ephemeris " means a table of coordinates of a celestial body given
for equidistant intervals of time.
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 63
Noon," and contains the following data for that instant for each
day in the month: right ascension and declination of the sun
and their hourly changes, sun's semidiameter, time of semi-
diameter passing the meridian, and the equation of time with
its hourly change. The second page is headed "At Greenwich
Mean Noon," and contains the right ascension and declination
of the sun with their hourly changes, the equation of time with
its hourly change, and the right ascension of the mean sun or
sidereal time at mean noon. These pages are the ones to be',
used by the navigator or the surveyor when making observa-(
tions on the sun. Whether Table I or Table II shall be used in
any given case depends upon whether apparent time or mean
time is the more convenient. If the declination, right ascension,
or equation of time is required for the instant of the sun's transit
over any meridian, then the local apparent time is noon and the
Greenwich Apparent Time is equal to the west longitude of the
place. The desired quantity is found by taking out its value
for the instant of Greenwich Apparent Noon and increasing or
decreasing it by the hourly change. multiplied by the number of
hours in the longitude. If the quantity is to be found for some
instant of local mean time or Standard Time, then the Greenwich
Mean Time may be readily found, and it is therefore more con-
venient to compute the required value from that given for Green-
wich Mean Noon. If Local Time is used, the Greenwich Time
is found by adding the longitude; if Standard Time is used, the
Greenwich Time is found by adding 5^, 6 h , etc., according to
the time belt indicated. The tabular quantity is then corrected
for the time elapsed since Greenwich Mean Noon. Tables I and
II for the month of January, 1912, are shown on pages 64 and 65.
The third page of the Almanac contains data not usually required
by the surveyor. On the fourth page are the semidiameter,
horizontal parallax, and time of transit of the moon, with their
hourly changes. On account of the rapidity with which the
semidiameter and parallax vary, they are given for both Green-
wich noon and midnight. The next eight pages contain the
AT GREENWICH APPARENT NOON
;al time ol
ed to ap-
h m s
18 42 21.37
S. 23 5 43.7
1 8 46 46.49
i i . 039
23 i i-5
18 51 11.26
22 55 51.6
18 55 35-68
22 50 14-4
18 59 59-71
22 44 IO.O
19 4 23-33
22 37 38.5
19 8 46.53
22 30 40-3
19 13 9-28
22 23 15-3
19 17 31-55
22 IS 23.9
19 21 53-31
22 7 6.3
19 26 14-54
21 58 22.7
19 30 35-23
21 49 13.3
19 34 55-33
21 39 38.4
19 39 14-83
21 29 38.3
19 43 33-71
21 19 I3.I
19 47 51-96
21 8 23.3
19 52 9-54
20 57 9.2
19 56 26.42
20 45 31.0
20 o 42.58
20 33 29.1
20 4 58.02
20 21 3-9
20 9 12.71
20 8 15.7
20 13 26.64
19 55 4-8
20 17 39-8o
19 41 31.7
20 21 52.16
19 27 36.7
12 1. 80
2O 26 3.72
19 13 20.0
2O 30 14.46
18 58 42.3
20 34 24.38
18 43 43-9
20 38 33-48
18 28 25.0
20 42 41-74
18 12 46.2
16 16. 16
2O 46 49.16
17 56 48.0
20 50 55.74
17 40 30.7
20 S5 1-49
S. 17 23 54.5
Note. The mean time of semidiameter passing may be found by subtracting o*.ig from the
sidereal time. The sign + prefixed to the hourly change of declination indicates that south decli-
nations are decreasing.
AT GREEN'WICH MEAN NOON
Day of the month.
of time, to
time, or right
h m s
18 42 20.78
18 46 45-81
18 51 10.50
i i . 049
S. 23 5 44.3
23 I 2.2
22 55 52-5
+ 11. 18
h m s
18 39 7-54
18 43 4. 10
18 47 0.66
18 55 34.84
18 59 58.79
19 4 22.33
22 50 15.5
22 44 11.3
22 37 40.1
18 50 57.22
18 54 53-78
18 58 50.33
19 8 45-45
19 13 8.12
19 17 30.30
22 30 42.0
22 23 17.3
22 15 26.2
19 2 46.89
19 6 43.45
19 10 40.01
19 21 51.99
19 26 13.15
19 3 33-77
22 7 8.8
21 58 25.5
21 49 16.4
19 14 36.57
19 18 33-13
19 22 29.68
19 34 53- 81
19 39 13-25
19 43 32.07
21 39 41.8
21 29 42.0
21 19 17.2
19 26 26.24
19 30 22.8o
19 34 19-36
19 47 50-25
19 52 7-76
19 56 24.58
21 8 27.7
20 57 13-9
20 45 36.1
19 38 15-92
19 42 12.48
19 46 9.03
20 o 40.70
20 4 56.09
20 9 10.73
20 33 34.6
2O 21 9.7
20 8 21.8
19 50 5-59
19 54 2.15
19 57 58-71
20 13 24.62
20 I? 37.74
20 21 50.06
19 55 n-2
19 41 38.4
19 27 43.7
20 i 55.27
20 5 51.82
20 9 48.38
20 26 1.58
2O 3O 12.29
20 34 22.18
19 13 27.4
18 58 50.0
18 43 51.9
20 13 44-94
20 17 41.50
20 21 38.05
20 38 31.25
20 42 39.48
20 46 46.87
2 So 53-44
18 28 33.4
18 12 55.0
i? 56 57-o
17 40 39.9
20 25 34 -61
20 29 31.17
20 33 27.72
20 3? 24.28
20 54 59-17
S. 17 24 4.0
20 41 20.84
Note. The semidiameter for mean noon may be assumed the same as that for ap-
parent noon. The sign 4- prefixed to the hourly change of declination indicates that
>outh declinations are decreasing.
Diff. for i
+9 8 -8s6s.
MEAN PLACES OF STARS, 1912.
WASHINGTON, JANUARY I<*.OO6.
Name of star.
v/ 33 Piscium
h m s
o o 40.808
o ' //
6 II en. 4?
-f- 20 176
. a Andromedae (Alphe-
o 3 co 162
4-28 3.6 16 t;o
v ft Cassiopeiae
O A 28 CO4.
4- ?8 3O "CT O7
O 4 ^6.83.2
46 i3 ?8.o6
2 2 Andromedae
O C 44. C7I
4-4C 34 C7 27
o 8 42 161
+ "3 0861
4- IA 4.1 3O 7?
* O *J
4~ 20 02 1
Oil 4? 6l 1
4- -26 17 CO C2
O 14. ?6 674
o i 8 42 04
O 1C 2O 74?
1 lAO 1 ?
6c 23 2O 80
O 2O ? ? 4.64
+1 27 8 CO
o 21 8 620
+ 5 2O4.3
77 A A CO 48
4- 2O 27O
O 21 ?6 2 ?4
42 47 I OC
1 2 Ceti
o 25 32 885
4. 26 36 2C
I? Ceti t
O 3.2 3 748
4- C3 24 J.C 87
TT Andromedae. . .
4- 3 TO7O
O ?? 1 \4.I'?I
4-28 50 2 68
O ^4 'I?. I 2.8
4- 3O 22 4.C 08
a Cassiop. (Schedir}.-\
o 35 30-339
+ 56 3 17-55
2 . 2
O 3Q IO. ^8l
+ 3 0126
18 28 9 82
4- 10 7oc
o Cassiopeiae ....
21 Cassiopeise .
17 Cassiopeise "\
O J."? d.6. 12^
-J- C7 2O CO C4
4- TO 63T
X Hydri .
o J.8 "?o cr^A
T 77 18 A7
O Cl 27 2J.7
4- OO T/L 2C C6
o r T c T Rrc
O CJ. 21 Ol8
+ 2 8908
43 H. Cephei
O ^6 71 2J.4
4- 8 c 4.7 81?
ft Phcenicis .... f
H Cassiopeiae. . .
T J. O *77C
ft Andr~ xlae
i 6 48 60*5
f Piscium f
T O 7 O?8
K Tucanae f
T T3 T C C2C
T TO 77 ACC
8 78 17 87
j.7 46 8 c8
I 2J. 7O 74O
+ 69 48 43 86
I 26 J.6 7O7
T 8 623
a Ursae Min.(Po/am)f
i 27 51.07*
+ 88 50 10.77
13 Ceti. dup. 5 w .s. 6*.2, o*.3.
a Cassiop.. var. irreg 2"*.2,2".8
17 Cassiop.. comp.7"*.6, 5" s. pr.
/3 Phoenicis, dup. 4 m .i,4 m .i,i*.
Piscium, star 6 m .s, 24" n.f.
K Tucanae, comp. 7 m , 6" n.
a TJrsse Min., star g m , 18" s. pr.
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 67
APPARENT PLACES OF STARS, 1912.
FOR THE UPPER TRANSIT AT WASHINGTON.
- 6 ii
+ 28 36
+ 58 39
26-57 3 '
65-1 t l
56.48 9 ,
62 '3 o
Feb. 9 . i
' ' 19
57-9 a .
55-4 2 J
Mar. 10. o
6 5-9 27
6 5-7 I
ii. 5 4
5- 1 2
63.0 2 '
2 5-5 c
56.01 x 9
Apr. 9 . o
9- 2 f.
J > 29
27-07 4 g
o 4 '
51.21 3 *
15-6 2 '
58 -7 6 !!
J 7' 8 !
51,93 2 g
52.19 3 !
Aug. 6 . 6
52.71 , 4
27- 8 2 <
-j 2 , 34
5 6 ' 2 L
60. 61 ,
s 2 -^ ;
Nov. 4 . 4
^ i 74
5o . QO> "i!
51.2 J 9
85.7 ;. 7
60. . 7
52.98 * o
60. >J 2 9
Sec 8, Tan 5
i . 006 o 109
i. 139 +o. 545
I . 446 I. 044
49 s - 898 59". 45
SO 8 . 162 16". 59
288.504 51". 97
5 6 8 .832 58". 96
O . OO . 04
. 00 . 1 1
+ 0.4 o.o
+ 0.4 o.o
+ 0.4 o.o
+ 0.4 o.o
68 PRACTICAL ASTRONOMY
moon's right ascension and declination for every hour of Green-
wich Mean Time, together with the changes per minute of time.
Following the ephemeris of the sun and moon for the twelve
months are the ephemerides of the planets. The values given
for Greenwich Mean Noon are for o h of the astronomical date
or 12 M of the civil date. The word " apparent " used in these
tables indicates that the correction for aberration has been
applied to the coordinates, giving the position of the object as
actually seen by the observer except for the effect of parallax
and refraction. The " differences for i hour " are in all cases
the rates of change at the instant, that is, the differential coeffi-
cients, not the actual differences between the consecutive tabular
Part II contains the following three lists of stars, the first
headed " Mean places of Stars, 19 "; the second, "Apparent
Places of Circumpolar Stars, 19 "; and the third, "Apparent
Places of Stars, 19 "; all of these are computed for the instant
of transit over the meridian of Washington (5* o8 TO 15*. 78 West
of Greenwich). A list of south circumpolar stars is also given.
The tables given on pages 66 and 67 of this volume are extracts
from the first and third of these tables in the Almanac. The
first table contains the coordinates of about 800 stars referred to
the "mean equinox " at the beginning of the year, that is, to
the position that the equinox would occupy at the beginning
of the solar * year if it were not affected by small periodic terms
of the precession. The second table gives the coordinates of
about 1 5 north circumpolar stars. Precession causes the coordi-
nates of circumpolar stars to vary more rapidly than those of
equatorial stars; the coordinates are therefore given for every
day in the year. The hours and minutes of right ascension and
the degrees and minutes of declination are at the head of the
column ; the column contains only the seconds. In the third table
* The year here referred to, called also the Besselian fictitious year, is one used
in computing star places; it begins when the sun's mean longitude is 280, that is,
when the R. A. of mean sun is i8 h 40"*, which occurs about Jan. i.
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 69
are about 800 stars, the coordinates for which are given for every
ten days. The only other table in Part II of particular interest to
the surveyor is that headed " Moon Culminations." This table
contains the data needed when determining longitude by observ-
ing transits of the moon. (See Art. 88, p. 141.)
In the latter part of the Almanac will be found the following
useful tables: I, Times of Culmination and Elongation of Polaris;
II, Conversion of Sidereal Time into Mean Time; III, Conver-
sion of Mean Time into Sidereal Time; IV, Latitude by an
Altitude of Polaris; V and VI, Azimuths of Polaris; VII, Intervals
for 5 Cassiopeia and f Ursa Majoris. (See Art. 99, p. 161.)
39. Star Catalogues.
When it is necessary to make observations on stars not given
in the list in the Ephemeris, their positions must be taken from
one or more of the star catalogues. These give the mean place
of the star at some definite epoch, such as the beginning of the
year 1890, or 1900, together with the necessary data for reducing
to the mean place of any other year. This data is usually
obtained by combining the observations made at different obser-
vatories and at different times, so that changes in the star's
coordinates are accurately determined. After the position in
the catalogue has been brought up to the mean place for the
desired year, the apparent place of the star for the exact date of
the observation is computed by means of the formulae and tables
given in Part II of the Ephemeris. For ordinary observations
made by the surveyor the list of stars given in the Ephemeris is
always sufficient, but in special kinds of work, such as finding
latitude by Talcott's Method, many other stars must be used.
When taking data from the Ephemeris it is general necessary,
in order to obtain the value for a particular instant, to interpolate
between values of the function for stated times. In some cases
this may be simple interpolation, in which the function is assumed
to vary uniformly between the two values given and the desired
value found by direct proportion. When the difference for one
70 PRACTICAL ASTRONOMY
hour is given, this rate of change at the given instant may be
assumed to hold good between the given value and the following
one. Since this is not usually quite true, it will be more accurate
to interpolate from the nearest given value in the table. The
change for one hour is to be multiplied by the number of hours
between the given time and the tabular time. This correction
is either added or subtracted, according to whether the function
is increasing or decreasing and whether the preceding or follow-
ing tabular value is used.
At Greenwich Mean Noon.
Feb. Sun's declination Diff. i h
1 Si7 2 4 'o 4 ".o + 4 i".88
2 17 07 09 .8 42 .64
It is desired to find the declination at the instant 22* G. M. T.
Feb. i. Since this is much nearer to the moon of Feb. 2 than
of Feb. i, it will be more accurate to multiply 42". 64 by 2 h and
add this to 17 07' 09". 8 (since the declination is decreasing).
The result is S 17 08' 35".!. By working forward from the
value on Feb. i the result is S 17 08' 3 2" . 6. By using a more
exact formula the result is found to be S 17 08' 35".o.
If the successive values of the " diff. for i h " or " diff. for
i m " have large differences, and if a precise value of the function
is desired, it will be necessary to interpolate between the given
values of the differential coefficients to obtain the rate of change
at the middle of the interval over which we are interpolating,
and to use this interpolated rate of change in computing the
Time R. A. of the Moon . Diff. for I OT
o* 4 ft 46 m n a .49 2 s . 54 2 1
i 4 48 44.06 2.5436
If it is desired to find the right ascension for o h 40, the " diff.
for i m " to be used is that for the instant o h 20, the middle
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 71
of the interval from o h to o h 40. This value lies one third of
the way from 2 s . 5421 to 2 S .5436, or 2 S .5426. The correction to
the R. A. at o h is 2 S .5426 X 4O TO = 101.70* = i m 4i s .7o, the
required R. A. being 4* 47"* 53 s . 19.
For general interpolation formulae the student is referred to
Chauvenet's Spherical and Practical Astronomy, Vol. I, to Doo-
little's Practical Astronomy or to Hayford's Geodetic Astronomy.
Questions and Problems
1. Compute the sun's apparent declination when the M. L. T. is 8 h 30 A.M.,
Jan. 16, 1912, at a place 85 west of Greenwich (see p. 65).
2. Compute the right ascension of the mean sun at local mean noon Jan. 10,
1912, at a place 96 10' west of Greenwich.
3. Compute the equation of time for local apparent noon Jan. 30, 1912, at a
place 20 east of Greenwich.
4. Explain the relation between the sun's angular semidiameter and the time
of the semidiameter passing the meridian.
5. What is the relation between the " right ascension of the mean sun " and
the " apparent right ascension of the sun " on Jan. i, 1912?
THE EARTH'S FIGURE CORRECTIONS TO OBSERVED
41. The Earth's Figure.
The earth's form is approximately that of an oblate spheroid
whose shortest axis is the axis of rotation. The actual figure
deviates slightly from that of a perfect spheroid, but for most
astronomical purposes these deviations may be disregarded.
Each meridian may therefore be considered as an ellipse, and
the equator and all parallels of latitude as circles. The semi-
major axis of the meridian ellipse is about 3962.80 miles, and the
semi-minor axis is 3949.56 miles in length. The length of i of
latitude at the equator is 68.704 miles; at the pole it is 69.407
In locating points on the earth's surface by means of coordi-
nates there are three kinds of latitude to be considered. The
latitude as found by astronomical observation is dependent upon
the direction of gravity as indicated by the spirit levels of the
instrument, and is affected by any abnormal deviations of the
plumb line* at this point; the latitude as found directly by ob-
servations is called the astronomical latitude. The geodetic
latitude is the latitude that would be found by observation if
the plumb line were normal to the surface of the spheroid taken
to represent the earth's figure, that is, if all of the irregularities
of the surface were smoothed out. Evidently the geodetic
latitude cannot be directly observed but must be found by com-
putation. The geocentric latitude is the angle between the plane
of the equator and a line drawn from the centre of the earth to
the point on the surface. In Fig. 39 the line AD is normal to
* These deviations are small, averaging about 3" or 4", but in some cases
deviations of nearly 30" are found.
THE EARTH'S FIGURE
the earth's surface at A, and the angle ABE is the geodetic
latitude of A. If the plumb line coincides with AD, this is
also the astronomical latitude. The angle ACE is the geocen-
tric latitude. The difference between the two, or angle BAC,
is called the angle of the vertical, or the reduction of latitude.
The geocentric latitude is always less than the observed lati-
tude by an angle which varies from about o n' 30" in latitude
45 to zero at the equator and the poles. Whenever observations
are made at any point on the earth's surface it is necessary to
reduce the observed values to their values at the earth's centre
before they can be combined with other data referred to the
centre. In making this reduction the geocentric latitude must
be used if the exact position of the observer with reference to the
centre is to be computed. For most of the observations treated
in the following chapters it will not be necessary to consider the
spheroidal shape of the earth; it will be sufficiently exact to regard
it as a sphere.
The coordinates of a celestial object as given in the Ephemeris
are referred to the centre of the earth, while the coordinates
obtained by observation are necessarily measured from a point
on the surface, and must be reduced to the centre. The case
most frequently occurring in practice is that in which the altitude
of an object is observed and the geocentric altitude is desired.
For all objects except the moon the distance of the body is so
great that it is sufficiently accurate to regard the earth as a
sphere. In Fig. 40, the angle ZOS is the observed zenith dis-
tance, or the complement of the observed altitude, and ZCS is
the true zenith distance. This apparent displacement of the
object on the celestial sphere is called parallax. The effect of
parallax is simply to decrease the altitude without altering the
azimuth of the body, provided the spheroidal form of the earth
be disregarded. The difference in direction between the lines
OS and CS, or the angle OSC, is the parallax correction. In the
triangle OSC, angle COS may be considered as known, since the
altitude or complement of ZOS is observed. The distance OC
is the semidiameter of the earth (3956.1 miles), and CS is the
THE EARTH'S FIGURE' 75
distance from the earth's centre to the centre of the body ob-
served. Solving this triangle,
sinS = smZOSX~ 
It is evident that the parallax correction will be zero at the
zenith and a maximum at the horizon. For the maximum,
when ZOS = 90,
. e OC r ,
sin 5 = > 
which is the same for all places on the earth's surface if the earth
is regarded as a sphere. If P h represents the maximum or
horizontal parallax, then equation  may be written
sin S = sin Ph sin z
= sin P h cos //, 
where h is the apparent altitude of the object. But S and Ph
are usually very small angles, and the error is negligible if the
sines are replaced by their arcs.* Equation  then becomes
S" = P h " cos h, 
where S" and P h " are both in seconds of arc.
For the moon the mean value of the horizontal parallax is
about o 57' 02" f; for the sun it is 8". 8; for the fixed stars it is
* The sine may be expressed as a series as follows:
x 3 x 5
sin x = x Y + T~ ~ ' ' ' 
Replacing sin x by x amounts to neglecting all terms after the first. Whether the
error will be appreciable in any given case may be determined by computing the
value of the first of the neglected terms. If x = i the neglected terms are less
than .005 of i% of x. The error in an angle of i would be less than o".2. The
moon is the only object whose parallax is nearly as large as i, so that for all other
objects this approximation is usually allowable. Similarly for cos x = i, the terms
neglected are those of the series
' I 
t The moon's mean distance is 238,800 miles; the sun's mean distance is
too small to be detected. The horizontal parallaxes of objects
in the solar system are given in the Nautical Almanac.* For the
parallax of the sun for different altitudes see Table IV (A).
Refraction is the term applied to the bending of a ray of light
by the atmosphere as it passes from a celestial object to the
observer's eye. On account of the increasing density of the
layers of air the rays of light coming from any object are bent
downward into a curve, and consequently when the rays enter
the eye they have a greater inclination to the horizon than they
did before entering the atmosphere. For this reason all objects
appear higher above the horizon than they actually are. In
Fig. 41, S is the true position of a star and 5" its apparent position.
The light from 5 is bent into a curve aO, and the star is seen in
the direction of the tangent ObS'. The angle which must be
subtracted from the altitude of S f to obtain the altitude of 6 1 is
called the refraction correction. This angle is really the angle
SOS', but on account of the great distance of celestial objects
* On account of the spheroidal form of the earth the equatorial diameter is the
greatest and the parallax at the equator is a maximum; the parallaxes are therefore
given in the Ephemeris under the heading " Equatorial Horizontal Parallax."
THE EARTH'S FIGURE 77
and the small angle of refraction the correction may be con-
sidered as the angle SbS'. From the figure it is evident that
ZcS = ZOS' + S'bS,
or z' = z + r, 
where z' = the true and z = the apparent zenith distance and
r = the refraction correction. The approximate law of astro-
nomical refraction may be deduced by assuming that the bend-
ing all occurs at point b. The general law of refraction, when a
ray enters a refracting medium, is expressed by the equation
sinz' = n sin z, 
where n is the index of refraction of the given medium; for air
its value is roughly about 1.0003.
Substituting from Equa. ,
sin (z + r) = n sin z, 
Expanding, sin z cos r + cos z sin r = n sin z. 
Since r is a small angle (never greater than 40') it is allowable
to put cos r = i and sin r = r; then
sin z + r cos z = n sin z,
and r cos z = (n i) sin z,
or r = (n i) tan z. 
Replacing n by 1.0003 an d dividing by sin i" to reduce r from
circular measure to seconds of arc,
r" = - ^- tan 2
= 60" tan z
= 60" cot A, 
where /t is the apparent altitude.*
The value of n varies considerably with the temperature and
the pressure of the air, so that equation  must be considered
as giving only a rough approximation to the true refraction.
* " Apparent" is used here simply to distinguish between the direction of the star
as actually seen and the direction unaffected by refraction. In speaking of parallax,
the word " apparent " has a different meaning, and in case of aberration, still
7 PRACTICAL ASTRONOMY
For high altitudes this formula is nearly correct, but for altitudes
under 10 it is not sufficiently exact. If both sides of the equa-
tion are divided by 60 so that r is reduced to minutes, we have
the extremely simple relation that the refraction in minutes equals
the natural cotangent of the altitude. For altitudes measured
with an engineer's transit this formula is close enough for alti-
tudes greater than about 10. For more accurate values of the
refraction Table I may be used. From the table it will be seen
that the refraction correction is zero at the zenith, about i' at
an altitude of 45, and about 6 34' at the horizon.*
The following formula, due to Professor George C. Comstock,
gives very accurate values of the refraction for altitudes greater
than 20, and is sufficiently accurate for all field observations
made with surveyors' instruments.
983 b , r , -,
r = I cot h, 
460 + t
in which b is the barometer reading in inches, and / is the tem-
perature in Fahrenheit degrees.
Altitude 30, barometer 29. i m ', thermometer 81 F.
log. 983 = 2.9926
log. 29.1 = 1.4639
460 colog. 541 = 7.2668
81 cot h = 0.2386
r = 9 i".6
= i' 3 i".6
The discs of the sun and moon are circular, and their angular
semidiameters are given for each day in the Ephemeris. Since
measurements can only be taken to the edge, or limb, the altitude
of the centre of the object is obtained by making a correction
* The sun's diameter is about 3 2', slightly less than the refraction on the horizon;
when the sun has actually gone below the horizon at sunset the entire disc is still
visible on account of the 34' increase in its apparent altitude due to atmospheric
THE EARTH'S FIGURE
equal to the semidiameter. The apparent angular semidiameters
given in the Ephemeris may be affected in two ways, one by the
change in the observer's distance because he is on the earth's
surface, the other by the difference in the amount of refraction
correction on the upper and lower edges of the disc.
The semidiameter given in the Ephemeris is that as seen from
the centre of the earth. When the object is in the zenith the
observer is nearly 4000 miles nearer than when it is in the hori-
zon. The moon is about 240,000 miles distant from the earth,
so that the semidiameter is increased by about ^V part, or
The vertical diameter of an object appears to be less than its
horizontal diameter because the refraction lifts the lower edge
more than it does the upper edge. The disc then presents the
appearance of an ellipse. When the sun is rising or setting, the
contraction is most noticeable. This contraction of the semi-
diameter does not affect the correction to an observed altitude,
but must be taken into account when the distance is measured
between the moon's limb and a star or a planet. (See Art. 108.)
For the angular semidiameter of
the sun on the first day of each
month see Table IV (B).
If altitudes are taken from the
sea horizon, as when observing
on board ship with the sextant,
the measured altitude must be
diminished by the angular dip
of the sea horizon below the true
horizon. In Fig. 42 suppose the
observer to be at 0; the true
horizon is OB and the sea horizon
OH. Let OP = h, the height in
feet above the surface; PC = R, the radius of the earth; and
D, the angle of dip.
80 PRACTICAL ASTRONOMY
Then cos Z> = -^-j-^ 
Putting cos D = i , neglecting other terms in the series,
D 2 h h , , v
R + h R
Replacing R by its value in feet, 20,884,000, and dividing by
sin i' to reduce D to minutes,
- X sin i'
This shows the amount of dip unaffected by refraction. The
effect of refraction is to apparently lift the horizon, and the dip
affecting the observed altitude is therefore less than that given
by the formula. If the coefficient 1.064 is taken as unity, the
formula is nearer the truth and is simpler, although still some-
what too large. Table IV (C) , based on a more exact formula, will
be seen to give smaller values. For ordinary sextant observations
made at sea, where the greatest precision is not required it is
sufficient to take the dip in minutes equal to the square root of
the height of the eye in feet, that is,
D' = Vh ft. 
46. Sequence of Corrections.
Strictly speaking, the corrections to the latitude should be
made in the following order :
(i) Instrumental corrections; (2) dip (if at sea); (3) refraction;
(4) semidiameter; (5) parallax. In practice, however, it is not
always necessary to follow this order exactly. At sea the cor-
rections are often taken together as a single " correction to the
altitude." Care should be taken to use the refraction correction
THE EARTH'S FIGURE 8 1
for the limb observed, not for the centre, for if the altitude is
small the two will differ appreciably.
1. Compute the sun's mean horizontal parallax. The sun's mean distance is
92,900,000 miles; for the earth's radius see Art. 41. Compute the sun's parallax
at an altitude of 60.
2. Compute the moon's mean horizontal parallax. The moon's mean distance
is 238,800 miles; for the earth's radius see Art. 41. Compute the moon's parallax
at an altitude of 45,
DESCRIPTION OF INSTRUMENTS
47. The Engineer's Transit.
The engineer's transit is an instrument for measuring hori-
zontal and vertical angles. For the purpose of discussing the
theory of the instrument it may be regarded as a telescopic line
of sight having motion about two axes at right angles to each
other, one vertical, the other horizontal. The line of sight is
determined by the optical centre of the object glass and the
intersection of two cross hairs* placed in its principal focus.
The vertical axis of the instrument coincides with the axes of
two spindles, one inside the other, each of which is attached to a
horizontal circular plate. The lower plate carries a graduated
circle for measuring horizontal angles; the upper plate has two
verniers, on opposite sides, for reading angles on the circle.
On the top of the upper plate are two uprights, or standards,
supporting the horizontal axis to which the telescope is attached
and about which it rotates. At one end of the horizontal axis
is a vertical arc, or a circle, and on the standard is a vernier, in
contact with the circle, for reading the angles. The plates and
the horizontal axis are provided with clamps and slow-motion
screws to control the motion. On the upper plate are two spirit
levels for levelling the instrument, or, in other words, for making
the vertical axis coincide with the direction of gravity.
The whole instrument may be made to turn in a horizontal
plane by a motion about the vertical axis, and the telescope may
be made to move in a vertical plane by a motion about the
horizontal axis. By means of a combination of these two
* Also called wires or threads; they are either made of spider threads or are
lines ruled upon glass.
DESCRIPTION OF INSTRUMENTS 83
motions, vertical and horizontal, the line of sight may be made
to point in any desired direction. The motion of the line of
sight in a horizontal plane is measured by the angle passed over
by the index of the vernier along the graduated horizontal
circle. The angular motion in a vertical plane is measured by
the angle on the vertical arc indicated by the vernier attached
to the standard. The direction of the horizon is denned by
means of a long spirit level attached to the telescope. When
the bubble is central the line of sight should lie in the plane of
the horizon. To be in perfect adjustment, (i) the axis of each
spirit level * should be in a plane at right angles to the vertical
axis; (2) the horizontal axis should be at right angles to the
vertical axis; (3) the line of sight should be at right angles to the
horizontal axis; (4) the axis of the telescope level should be
parallel to the line of sight, and (5) the vernier of the vertical
arc should read zero when the bubble is in the centre of the level
tube attached to the telescope. When the plate levels are
brought to the centres of their tubes, and the lower plate is so
turned that the vernier reads o when the telescope points south,
then the vernier readings of the horizontal plate and the vertical
arc for- any position of the telescope are coordinates of the
horizon system (Art. 12). If the horizontal circles are clamped
in any position and the telescope is moved through a complete
revolution, the line of sight describes a vertical circle on the
celestial sphere. If the telescope is clamped at any altitude and
the instrument turned about the vertical axis, the line of sight
describes a cone and traces out on the sphere a circle of equal
altitudes, or an almucantar.
48. Elimination of Errors.
It is usually more difficult to measure an altitude accurately
with the transit than to measure a horizontal angle. While the
precision of horizontal angles may be increased by means of
repetitions, in measuring altitudes the precision cannot be
* The axis of a level may be defined as a line tangent to the curve of the glass
tube at the point on the scale taken as the zero point, or at the centre of the tube.
84 PRACTICAL ASTRONOMY
increased by repeating the angles, owing to the construction of
the instrument. The vertical arc usually has but one vernier,
so that the eccentricity cannot be eliminated, and this vernier
often does not read as closely as the horizontal vernier. One
of the errors, which is likely to be large, but which may be elimi-
nated readily, is that known as the index error. The measured
altitude of an object may differ from the true reading for two
reasons: first, the zero of the vernier may not coincide with the
zero of the circle when the telescope bubble is in the centre of
its tube; second, the line of sight may not be horizontal when
the bubble is in the centre of the tube. The first part of this
error can be corrected by simply noting the vernier reading when
the bubble is central, and applying this as a correction to the
measured altitude. To eliminate the second part of the error
the altitude may be measured twice, once from the point on the
horizon directly beneath the object observed, and again from
the opposite point of the horizon. In other words, the instru-
ment may be reversed (180) about its vertical axis and the
vertical circle read in 'each position while the horizontal cross
hair of the telescope is sighting the object. The mean of the
two readings is free from the error in the sight line. Evidently
this method is practicable only with an instrument having a
complete vertical circle. If the reversal is made in this manner
the error due to non-adjustment of the vernier is eliminated at
the same time, so that it is unnecessary to make a special deter-
mination of it as described above. If the circle is graduated
in one direction, it will be necessary to subtract the second
reading from 180 and then take the mean between this result
and the first altitude. In the preceding description it is assumed
that the plate levels remain central during the reversal of the
instrument, indicating that the vertical axis is truly vertical.
If this is not the case, the instrument should be relevelled before
the second altitude is measured, the difference in the two altitude
readings in this case including all three errors. If it is not de-
sirable to relevel, the error of inclination of the vertical axis may
DESCRIPTION OF INSTRUMENTS 85
still be eliminated by reading the vernier of the vertical circle
in each of the two positions when the telescope bubble is central,
and applying these corrections separately. With an instru-
ment provided with a vertical arc only it is essential that the axis
of the telescope bubble be made parallel to the line of sight, and
that the vertical axis be made truly vertical. To make the axis
vertical without adjusting the levels themselves, bring both
bubbles to the centres of their tubes, turn the instrument 180
in azimuth, and then bring each bubble half way back to the
centre by means of the levelling screws. When the axis is truly
vertical, each bubble should remain in the same part of its tube
in all azimuths. The axis may always be made vertical by
means of the long bubble on the telescope; this is done by set-
ting it over one pair of levelling screws and centring it by means
of the tangent screw on the standard; the telescope is then
revolved about the vertical axis, and if the bubble moves from
the centre of its tube it is brought half way back by means of
the tangent screw, and then centred by means of the levelling
screws. This process should be repeated to test the accuracy
of the levelling; the telescope is then turned at right angles
to the first position and the whole process repeated. This
method should always be used when the greatest precision is
desired, because the telescope bubble is much more sensitive
than the plate bubbles.
If the line of sight is not at right angles to the horizontal axis,
or if the horizontal axis is not perpendicular to the vertical axis,
the errors due to these two causes may be eliminated by com-
bining two sets of measurements, one in each position of the
instrument. If a horizontal angle is measured with the vertical
circle on the observer's right, and the same angle again observed
with the circle on his left, the mean of these two angles is free
from both these errors, because the two positions of the horizontal
axis are placed symmetrically about a true horizontal line,* and
* Strictly speaking, they are placed symmetrically about a perpendicular to
the vertical axis.
86 PRACTICAL ASTRONOMY
the two directions of the sight line are situated symmetrically
about a true perpendicular to the rotation axis of the telescope.
If the horizontal axis is not perpendicular to the vertical axis the
line of sight describes a plane which is inclined to the true vertical
plane. In this case the sight line will not pass through the zenith,
and both horizontal and vertical angles will be in error. In
instruments intended for precise work a striding level is provided,
which may be set on the pivots of the horizontal axis. This
enables the observer to level the axis or to measure its inclina-
tion without reference to the plate bubbles. The striding level
should be used in both the direct and the reversed position and
the mean of the two results used in order to eliminate the errors
of adjustment of the striding level itself. If the line of sight is
not perpendicular to the horizontal axis it will describe a cone
whose axis is the horizontal axis of the instrument. The line
of sight will in general not pass through the zenith, even though
the. horizontal axis be in perfect adjustment. The instrument
must either be used in two positions, or else the cross hairs must
be adjusted. Except in large transits it is not usually practicable
to determine the amount of the error and allow for it.
49. Attachments to the Engineer's Transit. Reflector.
When making star observations with the transit it is necessary
to make some arrangement for illuminating the field of view.
Some transits are provided with a special shade tube into which
is fitted a mirror set at an angle of 45 and with the central
portion removed. By means of a lantern held at one side of
the telescope light is reflected down the tube. The cross hairs
appear as dark lines against the bright field. The stars can be
seen through the opening in the centre of the mirror. If no
special shade tube is provided, it is a simple matter to make a
substitute, either from a piece of bright tin or by fastening a
piece of tracing cloth or oiled paper over the objective. A hole
about | inch in diameter should be cut out, so that the light from
the star may enter the lens. If cloth or paper is used, the lan-
tern must be held so that the light is diffused in such a way as
DESCRIPTION OF INSTRUMENTS 87
tQ render the cross hairs visible. The light should be held so as
not to shine into the observer's eyes.
50. Prismatic Eyepiece.
When altitudes greater than about 55 to 60 are to be meas-
ured, it is necessary to attach to the eyepiece a totally reflecting
prism which reflects the rays at right angles to the sight line.
By means of this attachment altitudes as great as 75 can be
measured. In making observations on the sun it must be
remembered that the prism inverts the image, so that with a
transit having an erecting eyepiece with the prism attached the
apparent lower limb is the true upper limb; the positions of the
right and left limbs are not affected by the prism.
51. Sun Glass.
In making observations on the sun it is necessary to cover the
eyepiece with a piece of dark glass to protect the eye from the
sunlight while observing. The sun glass should not be placed
in front of the objective. If no shade is provided with the
instrument, sun observations may be made by holding a piece
of paper behind the eyepiece so that the sun's image is thrown
upon it. By drawing out the eyepiece tube and varying the
distance at which the paper is held, the images of the sun and
the cross hairs may be sharply focussed. By means of this
device an observation may be quite accurately made after a
52. The Portable Astronomical Transit.
The astronomical transit differs from the surveyor's transit chiefly in size and
in the manner of support. The diameter of the object glass may be anywhere
from 2 to 4 inches, and the focal length from 24 to 48 inches. The instrument is
set upon a stone or brick pier. The cross hairs usually consist of several vertical
hairs (say n or more) instead of a single one as in the surveyor's transit. The
motion in altitude is controlled by means of a clamp and a tangent screw. The
azimuth motion is usually very small, simply enough to allow adjustments to be
made, as the transit is not used for measuring horizontal angles. The axis is
levelled or its inclination measured by means of a sensitive striding level.
On account of the high precision of the work done with the astronomical transit
the various errors have to be determined with great accuracy, and corresponding
corrections applied to the observed results. The transit is chiefly used in the plane
of the meridian for determining the times of transit of stars. The principal errors
determined and allowed for are (i) azimuth, or deviation from the true meridian;
(2) inclination of the horizontal axis; (3) collimation, or deviation of the sight line
from the true perpendicular to the rotation axis. The corrections to reduce an
observed time to the true time of transit across the meridian are given by formulae
 to . These corrections would apply equally well to observations with the
engineer's transit, and serve to show the relative magnitudes of the errors for
different positions of the objects observed.
Azimuth correction = a cos h sec D, 
Level correction = b sin h sec D, 
Collimation correction = c sec D, 
where a, b and c are the errors in azimuth, inclination and collimation respectively
(expressed in seconds of time), and h is the altitude and D the declination of the
star observed. From these formulae Table B has been computed. It is assumed
that the instrument is i', or 4", out of the meridian (a = 4 s ); that the axis is
inclined i', or 4 s , to the horizon (b = 4*); and that the sight line denned by the
middle (or the mean) wire is i', or 4 s , to the right or left of its true position (c=4 8 ).
The numbers in the table show the effect of these errors at different altitudes and
TABLE B. ERROR IN OBSERVED TIME OF TRANSIT (IN
SECONDS OF TIME) WHERE a, b OR c = i'.
o s .o
O S .O
o s .o
o s .o
o s .o
I . I
Note. Use the bottom line for the collimation error.
53. The Sextant.
The sextant is an instrument for measuring the angular dis-
tance between two objects, the angle always lying in the plane
DESCRIPTION OF INSTRUMENTS
through the two objects and the eye of the observer. It is
particularly useful at sea because it does not require a steady
support like the transit. It consists of a frame carrying a
graduated arc, AB, Fig. 43, about 60 long, and two mirrors /
and H, the first one movable, the second one fixed. At the
center of the arc, 7, is a pivot on which swings an arm IV, 6 to
8 inches long. This arm carries a vernier V for reading the
angles on the arc AB. Upon this arm is placed the index glass
/. At H is the horizon glass. Both of these mirrors are set
so that their planes are perpendicular to the plane of the arc
AB, and so that when the vernier reads o the mirrors are parallel.
The half of the mirror H which is farthest from the frame is
unsilvered, so that objects may be viewed directly through the
glass. In the silvered portion other objects may be seen by
reflection from the mirror I to the mirror H and thence to
point O. At a point near (on the line HO) is a telescope of
low power for viewing the objects. Between the two mirrors
90 PRACTICAL ASTRONOMY
and also to the left of H are colored shade glasses to be used when
making observations on the sun. The principle of the instru-
ment is as follows : A ray of light coming from an object at
C is reflected by the mirror / to H, where it is again reflected
to O. The observer sees the image of C in apparent coincidence
with the object at D. The arc is so graduated that the reading
of the vernier gives directly the angle between OC and OD.
Drawing the perpendiculars FE and HE to the planes of the
two mirrors, it is seen that the angle between the mirrors is
a )8. Prolonging CI and DH to meet at O, it is seen that the
angle between the two objects is 2 a 2 /3. The angle between
the mirrors is therefore half the angle between the objects that
appear to coincide. In order that the true angle may be read
directly from the arc each half degree is numbered as though it
were a degree. It will be seen that the position of the vertex O
is variable, but since all objects observed are at great distances
the errors caused by changes in the position of are always
negligible in astronomical observations.
The sextant is in adjustment when, (i) both mirrors are per-
pendicular to the plane of the arc; (2) the line of sight of the
telescope is parallel to the plane of the arc; and (3) the vernier
reads o when the mirrors are parallel to each other. If the
vernier does not read o when the doubly reflected image of a
point coincides with the object as seen directly, the index cor-
rection may be determined and applied as follows. Set the
vernier to read about 30' and place the shades in position for
sun observations. When the sun is sighted through the tele-
scope two images will be seen with their edges nearly in contact.
This contact should be made as nearly perfect as possible and
the vernier reading recorded. This should be repeated several
times to increase the accuracy. Then set the vernier about 30'
on the opposite side of the zero point and repeat the whole
operation, the reflected image of the sun now being on the
opposite side of the direct image. If the shade glasses are of
different colors the contacts can be more precisely made. Half
DESCRIPTION OF INSTRUMENTS 91
the difference of the two (average) readings is the index correc-
tion. If the reading off the arc was the greater, the correction
is to be added to all readings of the vernier; if the greater reading
was on the arc, the correction must be subtracted,
In measuring an altitude of the sun above the sea horizon the
observer directs the telescope to the point on the horizon ver-
tically under the sun and then moves the index arm until the
reflected .image of the sun comes into view. The sea horizon
can be seen through the plain glass and the sun is seen in the
mirror. The sun's lower limb is then set in contact with the
horizon line. In order to be certain that the angle is measured
to the point vertically beneath the sun, the instrument is tipped
slowly right and left, causing the sun's image to describe an arc.
This arc should be just tangent to the horizon. If at any point
the sun's limb goes below the horizon the altitude measured is
too great. The vernier reading corrected for index error and
dip is the apparent altitude of the lower limb above the true
54. Artificial Horizon.
When altitudes are to be measured on land the visible horizon
cannot be used, and the artificial horizon must be used instead.
The surface of any heavy liquid, like mercury, molasses, or
heavy oil, may be used for this purpose. When the liquid is
placed in a basin and allowed to come to rest, the surface is
perfectly level, and in this surface the reflected image of the sun
may be seen, the image appearing as far below the horizon as
the sun is above it. Another convenient form of horizon con-
sists of a piece of black glass, with plane surfaces, mounted on a
frame supported by levelling screws. This horizon is brought
into position by placing a spirit level on the glass surface and
levelling alternately in two positions at right angles to each
other. This form of horizon is not as accurate as the mercury
surface but is often more convenient. The principle of the
artificial horizon may be seen from Fig. 44. Since the image
seen in the horizon is as far below the true horizon as the sun is
92 PRACTICAL ASTRONOMY
above it, the angle between the two is 2 h. In measuring this
angle the observer points his telescope toward the artificial
horizon and then brings the reflected sun down into the field of
view by means of the index arm. By placing the apparent
lower limb of the reflected sun in contact with the apparent
upper limb of the image seen in the mercury surface, the angle
measured is twice the altitude of the sun's lower limb. The two
points in contact are really images of the same point. If the
telescope inverts the image, this statement applies to the upper
limb. The index correction must be applied before the angle is
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.
The chronometer is simply an accurately constructed watch
with a special form of escapement. Chronometers may be
DESCRIPTION OF INSTRUMENTS 93
regulated for either sidereal or mean time. The beat is usually
a half second. Those designed to register the time on chrono-
graphs are arranged to break an electric circuit at the end of
every second or every two seconds. The 6oth second is dis-
tinguished either by the omission of the break at the previous
second, or by an extra break, according to the construction of the
chronometer. Chronometers are usually hung in gimbals to
keep them level at all times; this is invariably done when they
are taken to sea. It is important that the temperature of the
chronometer should be kept as nearly uniform as possible, be-
cause fluctuation in temperature is the greatest source of error.
Two chronometers of the same kind cannot be directly com-
pared with great accuracy, o s .i or o s .2 being about as close as
the difference can be estimated. But a sidereal and a solar chro-
nometer can easily be compared within a few hundredths of a
second. On account of the gain of the sidereal on the solar
chronometer, the beats of the two will coincide once in about
every 3 m 05*. If the two are compared at the instant when the
beats are apparently coincident, then it is only necessary to
note the seconds and half seconds, as there are no fractions to
be estimated. By making several comparisons and reducing
them to some common instant of time it is readily seen that
the comparison is correct within a few hundredths of a second.
The accuracy of the comparison depends upon the fact that the
ear can detect a much smaller interval between the two beats
than can possibly be estimated when comparing two chronome-
ters whose beats do not coincide.
The chronograph is an instrument for recording the time kept by a chronometer
and also any observations the times of which it is desired to determine. A piece
of paper is wrapped about a cylinder, which is revolved by a mechanism at a uniform
rate. A pen in contact with the paper is held on an arm, connected with the arma-
ture of an electro-magnet, in such a way that the pen draws a continuous line which
has notches in it corresponding to the breaks in the circuit made by the chro-
nometer. By means of this instrument the time is represented accurately on the
sheet as a linear distance. If it is desired to record the instant when any event
occurs, such as the passage of a star over a cross hair, the observer presses a tele-
graph key which breaks the same circuit, and a mark is made on the chronograph
sheet. The instant of the observation may be scaled from the record sheet with
57. The Zenith Telescope.
The zenith telescope is an instrument designed for making observations for
latitude by a special method devised by Capt. Andrew Talcott, and which bears
his name. The instrument consists of a telescope having a vertical and a horizontal
axis like the transit; the telescope is attached to one end of the horizontal axis in-
stead of at the centre. The essential features of the instrument are (i) a microm-
eter, placed in the focus of the eyepiece, for
^^ \ r^*+^ measuring small differences in zenith distance,
and (2) a sensitive spirit level, attached to a
small vertical circle on the telescope tube, for
measuring small deflections of the vertical axis.
The telescope is used in the plane of the
meridian. There are two stops whose positions
can be regulated so that the telescope may be
quickly shifted, by a rotation about the ver-
tical axis, from the north meridian to the
south meridian. The observation consists in
measuring with the micrometer the difference
in zenith distance of two stars, one north of
the zenith and one south, which culminate
- within a few minutes of each other, and in
taking readings of the spirit level at the same
time the micrometer settings are made. A
FIG. 45. THE ZENITH TELESCOPE diagram of the instrument in the two posi-
tions is given in Fig. 45. The inclination of
the telescope to the vertical is not changed between the two observations, so it is
essential that the zenith distances of the two stars should be so nearly equal that
both will come within the range of the micrometer screw, usually 30' or less.
The principle involved in this method may be seen from Fig. 46. From the
observed zenith distance of the star Ss the latitude is
and from the star
Taking the mean,
L = D n z n .
L = HA, + /?) + i (*. - z n ).
The latitude is therefore the mean of the declinations corrected by half the djffer-
ence of the zenith distances. The declination may be computed from the star
catalogues, and the difference in zenith distance may be very accurately measured
with the micrometer screw. It is evidently essential that the telescope should
DESCRIPTION OF INSTRUMENTS
have the same inclination to the vertical in each case. If the inclination changes,
however, the amount of this change is accurately determined from the level readings
already mentioned (see Art. 70).
58. Suggestions about Observing.
The instrument used for making such observations as are
described in this book will usually be either the engineer's transit
or the sextant. In using the transit care must be taken to give
the tripod a firm support. It is well to set the transit in position
some time before the observations are to be begun; this allows
the instrument to assume the temperature of the air and the
tripod legs to come to a firm bearing on the ground. The
observer should handle the instrument with great care, par-
ticularly during night observations, when the instrument is
likely to be accidentally disturbed. In reading angles at night
it is important to hold the light in such a position that the
graduations on the circle are plainly visible and may be viewed
along the lines of graduation, not obliquely. By changing the
position of the lantern and the position of the eye it will be
found that the reading varies by larger amounts than would be
expected when reading in the daylight. Care should be taken
not to touch the graduated silver circles, as they soon become
tarnished. The lantern should be held so as to heat the instru-
ment as little as possible, and so as not to shine into the observer's
eyes. Time may be saved and mistakes avoided if the program
of observations is laid out beforehand, so that the observer knows
just what is to be done and the proper order of the different
steps. The observations should be arranged so as to eliminate
instrumental errors, usually by means of reversals; but if this
is not practicable, then the instrument must be put in good
adjustment. The index correction should be determined and
applied, unless it can be eliminated by the method of observing.
In observations for time it will often be necessary to use an
ordinary watch. If there are two observers, one can read the
time while the other makes the observations. If a chronometer
is used, one observer may easily do the work of both, and at the
c sec. h
same time increase the accuracy. In making observations by
this method (called the " eye and ear method ") the observer
looks at the chronometer, notes the reading at some instant, say
at the beginning of some minute, and, listening to the half-second
beats, carries along the count mentally and without looking at
the chronometer. In this way he can note the second and
estimate the fraction without taking his attention from the star
and cross hair. After making his observation he may check his
count by again looking at the chronometer to see if the two
agree. After a little practice this method can be used easily
and accurately. In using a watch it is possible for one observer
to make the observations and also note the time, but it cannot
be done with any such precision as with the chronometer, be-
cause on account of the rapidity of the ticks (5 per second),
the observer cannot count the seconds mentally. The observer
DESCRIPTION OF INSTRUMENTS
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.
i. Show that if the sight line makes an angle c with the perpendicular to the
horizontal axis (Fig. 47) the horizontal angle between two points is in error by
c sec h' c sec h",
where h' and h" are the altitudes of the two points.
i tan h
2. Show that if the horizontal axis is inclined to the horizon by the angle *
(Fig. 48) the effect upon the azimuth of the sight line is i tan h, and that an angle
is in error by
i (tan h' - tan h"),
where &' and h" are the altitudes of the points.
59. The Constellations.
A study of the constellations is not really a part of the subject
of Practical Astronomy, and in much of the routine work of
observing it would be of comparatively little value, since the
stars used can be identified by means of their coordinates and a
knowledge of their positions in the constellations is not essential.
If an observer has placed his transit in the meridian and knows
approximately his latitude and the local time, he can identify
stars crossing the meridian by means of the times and the alti-
tudes at which they culminate. But in making occasional
observations with small instruments, and where much of the
astronomical data is not known to the observer at the time, some
knowledge of the stars is necessary. When a surveyor is be-
ginning a series of observations in a new place and has no accu-
rate knowledge of his position nor the position of the celestial
sphere at the moment, he must be able to identify certain stars
in order to make approximate determinations of the quantities
60. Method of Naming Stars.
The whole sky is divided in an arbitrary manner into irregular
areas, all of the stars in any one area being called a constellation
and given a special name. The individual stars in any constel-
lation are usually distinguished by a name, a Greek letter,* or
a number. The letters are usually assigned in the order of
brightness of the stars, a being the brightest, /3 the next, and so
on. A star is named by stating first its letter and then the name
of the constellation in the (Latin) genitive form. For instance,
* The Greek alphabet is given on p. 190.
THE CONSTELLATIONS 99
in the constellation Ursa Minor the star a is called a Ursa
Minoris; the star Vega in the constellation Lyra is called
a Lyra. When two stars are very close together and have
been given the same letter, they are often distinguished by the
numbers i, 2, etc., written above the letter, as, for example,
a 2 Capricorni, meaning that the star passes the meridian after
a 1 Capricorni.
* 61. Magnitudes.
The brightness of stars is shown on a numerical scale by their
magnitudes. A star having a magnitude i is brighter than one
having a magnitude 2. On the scale of magnitudes in use a few
of the brightest stars have fractional or negative magnitudes.
Stars of the fifth magnitude are visible to the naked eye only
under favorable conditions. Below the fifth magnitude a tele-
scope is usually necessary to render the star visible.
62. Constellations Near the Pole.
The stars of the greatest importance to the surveyor are those
near the pole. In the northern hemisphere the pole is marked
by a second-magnitude star, called the polestar, Polaris, or
a Ursa Minoris, which is about i 10' distant from the pole
at the present time (1910). This distance is now decreasing
at the rate of about one-third of a minute per year, so that for
several centuries this star will be close to the celestial north pole.
On the same side of the pole as Polaris, but much farther from
it, is a constellation called Cassiopeia, the five brightest stars
of which form a rather unsymmetrical letter W (Fig. 49). The
lower left-hand star of this constellation, the one at the bottom
of the first stroke of the W, is called 5, and is of importance to
the surveyor because it is very nearly on the hour circle passing
through Polaris and the pole; in other words its right ascension
is nearly the same as that of Polaris. On the opposite side of
the pole from Cassiopeia is Ursa Major, or the great dipper, a
rather conspicuous constellation. The star f , which is at the
bend in the dipper handle, is also nearly on the same hour circle
as Polaris and 5 Cassiopeia. If a line be drawn on the sphere
100 PRACTICAL ASTRONOMY
between 8 Cassiopeia and f Ursa Majoris, it will pass nearly
through Polaris and the pole, and will show at once the position
of Polaris in its diurnal circle. The two stars in the bowl of
the great dipper on the side farthest from the handle are in a
line which, if prolonged, would pass near to Polaris. These
stars are therefore called the pointers and may be used to find
the polestar. There is no other star near Polaris which is
likely to be confused with it. Another star which should be
remembered is /3 Cassiopeia, the one at the upper right-hand
corner of the W. Its right ascension is very nearly O A and
therefore the hour circle through it passes nearly through the
equinox. It is possible then, by simply glancing at Cassiopeia
and the polestar, to estimate approximately the local sidereal
time. When /3 Cassiopeia is vertically above the polestar it
is nearly O A sidereal time; when the star is below the polestar
it is 1 2 h sidereal time ; half way between these positions, left and
right, it is 6 h and iS h , respectively. In intermediate positions
the hour angle of the star ( = sidereal time) may be roughly
63. Constellations Near the Equator.
The principal constellations within 45 of the equator are
shown in Figs. 50 to 52. Hour circles are drawn for each hour
of R. A. and parallels for each 10 of declination. The approxi-
mate declination and right ascension of a star may be obtained
by scaling the coordinates from the chart. The position of the
ecliptic, or sun's path in the sky, is shown as a curved line. The
moon and the planets are always found near this circle because
the planes of their orbits have only a small inclination to the
earth's orbit. A belt extending about 8 each side of the ecliptic
is called the Zodiac, and all the members of the solar system
will always be found within this belt. The constellations along
this belt, and which have given the names to the twelve " signs
of the Zodiac," are Aries, Taurus, Gemini, Cancer, Leo, Virgo,
Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces.
These constellations were named many centuries ago, and the
FlG. 49. CONSTELLATIOJ
BOUT THE NORTH POLE
FIG. 50. PRINCIPAL FIXED STARS BETWEE:
:CLINATIONS 45 NORTH AND 45 SOUTH
FIG. 51. PRINCIPAL FIXED STARS BETWEI
DECLINATIONS 45 NORTH AND 45 SOUTH
FIG. 52. PRINCIPAL FIXED STARS BETWEI
)ECLINATIONS 45 NORTH AND 45 SOUTH
FlG. 53. CONSTELLATIO
ABOUT THE SOUTH POLE
THE CONSTELLATIONS IOI
names have been retained, both for the constellations themselves
and also for the positions in the ecliptic which they occupied at
that time. But on account of the continuous westward motion
of the equinox, the " signs " no longer correspond to the con-
stellations of the same name. For example, the sign of Aries
extends from the equinoctial point to a point on the ecliptic
30 eastward, but the constellation actually occupying this
space at present is Pisces. In Figs. 50 to 52 the constellations
are shown as seen by an observer on the earth, not as they would
appear on a celestial globe. On account of the form of pro-
jection used in these maps there is some distortion, but if the
observer faces south and holds the page up at an altitude equal
to his colatitude, the map represents the constellations very
nearly as they will appear to him. The portion of the map to be
used in any month is that marked with the name of the month
at the top; for example, the stars under the word " February "
are those passing the meridian in the middle of February at
about 9 P.M. For other hours in the evening the stars on the
meridian will be those at a corresponding distance right or left,
according as the time is earlier or later than 9 P.M. The approxi-
mate right ascension of a point on the meridian may be found at
any time as follows: First compute the R. A. of the sun by
allowing 2 h per month, or more nearly 4 per day for every
day since March 23, remembering that the R. A. of the sun is
always increasing. Add this R. A. to the local mean time and
the result is the sidereal time or right ascension of a star on the
Example. On October 10 the R. A. of the sun is 6 X 2 h + 17 X
4 TO = 13* o8 m . At g h P.M. (local mean time) the sidereal time
is 13* o8 w + g h oo m = 22 h oS m . A star having a R. A. of 22* o8 m
wgould therefore be close to the meridian at 9 P.M.
Fig. 53 shows the stars about the south celestial pole. There
is no bright star near the south pole, so that the convenient
methods of determining the meridian by observations on the
polestar are not practicable in the southern hemisphere.
102 PRACTICAL ASTRONOMY
64. The Planets.
In using the star maps, the student should be on the lookout
for planets. These cannot be placed on the maps because their
positions are rapidly changing. If a bright star is seen near the
ecliptic, and its position does not correspond to that of a star
on the map, it is a planet. The planet Venus is very bright and
is never very far from the sun; it will therefore be seen a little
before sunrise or a little after sunset. Mars, Jupiter, and Saturn
are outside the earth's orbit and therefore revolve around the
earth. Jupiter is the brightest, and when looked at through a
small telescope shows a disc like that of the full moon, and four
satellites can usually be seen all lying nearly in a straight line.
Saturn is not as large as Jupiter, but in a telescope of moderate
power its rings can be distinguished, or at least the planet looks
elongated. Mars is reddish in color and shows a disc.
OBSERVATIONS FOR LATITUDE
IN this chapter and the three immediately following are given
the more common methods of determining latitude, time, longi-
tude, and azimuth with small instruments. Those which are
simple and direct are printed in large type, and may be used for
a short course in the subject. Following these are given, in
smaller type, several methods which, although less simple, are very
useful to the engineer; these methods require a knowledge of
other data which the engineer must obtain by observation, and
are therefore better adapted to a more extended course of study.
65. Latitude by a Circumpolar Star at Culmination.
This method may be used with any circumpolar star, but
Polaris is the best one to use, when it is practicable to do so,
because it is of the second magnitude, while all of the other
close circumpolars are quite faint. The observation consists
in measuring the altitude of the star when it is a maximum or a
minimum, or, in other words, when it is on the observer's me-
ridian. This altitude may be obtained by trial, and it is not
necessary to know the exact instant when the star is on the
meridian. The approximate time when the star is at culmina-
tion may be obtained from Table V or by formulae  and .
It is not necessary to know the time with accuracy, but it will
save unnecessary waiting if the time is known approximately.
In the absence of any definite knowledge of the time of culmina-
tion, the position of the pole star with respect to the meridian may
be estimated by noting the positions of the constellations. When
S Cassiopeia is directly above or below Polaris the latter is at
upper or lower culmination. The observation should be begun
some time before one of these positions is reached. The hori-
104 PRACTICAL ASTRONOMY
zontal cross hair of the transit should be set on the star* and the
motion of the star followed by means of the tangent screw of the
horizontal axis. When the desired maximum or minimum, is
reached the vertical arc is read. The index correction should
then be determined. If the instrument has a complete vertical
circle and the time of culmination is known approximately, it
will be well to eliminate instrumental errors by taking a second
altitude with the instrument reversed, provided that neither
observation is made more than 4 m or 5 m from the time of culmi-
nation. If the star is a faint one, and therefore difficult to find,
it may be necessary to compute its approximate altitude (using
the best known value for the latitude) and set off this altitude
on the vertical arc. The star may be found by moving the
telescope slowly right and left until the star comes into the field
of view. Polaris can usually be found in this manner some time
before dark, when it cannot be seen with the unaided eye. It
is especially important to focus the telescope carefully before
attempting to find the star, for the slightest error of focus may
render the star invisible. The focus may be adjusted by look-
ing at a distant terrestrial object or, better still, by sighting at
the moon or at a planet if one is visible. If observations are to
be made frequently with a surveyor's transit, it is well to have
a reference mark scratched on the telescope tube, so that the
objective may be set at once at the proper focus.
The latitude is computed from Equa.  or . The true
altitude h is derived from the reading of the vertical circle by
applying the index correction with proper sign and then subtract-
ing the refraction correction (Table I). The polar distance is
found by taking from the Ephemeris (Table of Circumpolar
Stars) the apparent declination of the star and subtracting this
* The image of a star is practically a point of light; if the telescope were perfect
it would be actually a point, but, owing to the imperfections in the corrections
for aberration, the image, even though perfectly distinct, has an appreciable
width. The image of the star should be bisected with the horizontal cross hair.
OBSERVATIONS FOR LATITUDE 105
Observed altitude of Polaris at upper culmination = 43 37';
index correction = +30"; declination = +88 44' 35".
Vertical circle = 43 37' oo"
Index correction = +30
Observed altitude =43 37 30
Refraction correction = i 02
True altitude = 43 36 28
Polar distance = i 15 25
Latitude = 42 21' 03"
Since the vertical circle reads only to i' the resulting value for the
latitude must be considered as reliable only to the nearest i'.
Observed altitude of 51 Cephei at lower culmination = 39
33' 3"j index correction = o"; declination = -f- 87 n' 25".
Observed altitude = 39 33' 30"
Refraction correction = i 10
True altitude =39 32 20
Polar distance = 2 48 35
Latitude = 42 20' 55"
\A 66. Latitude by Altitude of Sun at Noon.
The altitude of the sun at noon (meridian passage) may be
determined by placing the line of sight of the transit in the plane
of the meridian and observing the altitude of the upper or lower
limb of the sun when it is on the vertical cross hair. The watch
time at which the sun will pass the meridian may be computed
by converting i2 h local apparent time into Standard or locaJ
mean time (whichever is used) as shown in Arts. 28 and 35.
Usually the direction of the meridian is not known, so the maxi-
mum altitude of the sun is observed and assumed to be the same
as the meridian altitude. On account of the sun's changing
declination the maximum altitude is not quite the same as the
meridian altitude; the difference is quite small, however, usually
a fraction of a second, and may be entirely neglected for obser-
vations made with the engineer's transit or the sextant. The
maximum altitude of the upper or lower limb is found by trial,
106 PRACTICAL ASTRONOMY
the horizontal cross hair being kept tangent to the limb as long
as it continues to rise. When the observed limb begins to drop
below the cross hair the altitude is read from the vertical arc
and the index correction is determined. The true altitude of
the centre of the sun is then found by applying the corrections for
index error, refraction, semidiameter, and parallax. In order
to compute the latitude it is necessary to know the sun's declina-
tion at the instant the altitude was taken. If the longitude of
the place is known approximately (say within half a degree) the
declination may be taken from the Nautical Almanac for the
instant of Greenwich Apparent Noon and increased or decreased
by the hourly change multiplied by the number of hours in the
longitude. If the place is west of Greenwich the correction is to
be added algebraically; if the place is east, it is to be subtracted.
If the longitude is not known, but the Greenwich mean time is
known, as would be the case if the timepiece kept either Green-
wich time or Standard time, the declination may be computed
by noting the watch time of the observation as nearly as possible
and correcting the declination as follows: take out the declina-
tion at Greenwich Mean Noon, and increase it by the hourly
change multiplied by the number of hours since Greenwich Mean
Noon. The latitude is then found from Equa. .
Observed maximum altitude of sun's lower limb, Jan. 8, 1906, = 25 06'; index
correction = +i'; the longitude is 4>44*i8 s (= 71 04' 5) west; the declina-
tion of the sun at Greenwich Apparent Noon = S 22 19' 33"; hourly change =
+ i9".59; the semidiameter = 16' 17".
Observed altitude = 25 06'. o Decl. at G. A. N. = - 22 19' 33"
Index correction + i .o 19". 59 X 4 A - 74 = +i 33
25 07 . o Decl. at L. A. N. = 22 18' oo"
Refraction correction = 2.0
25 05 . o
Semidiameter = + 16 .3
Altitude of centre
22 l8 . O
47 39'- 4
42 20'. 6
OBSERVATIONS FOR LATITUDE 107
Observed double altitude of sun's upper limb at noon on Jan. 28, 1910 (with
artificial horizon), = 59 17' 40"; Eastern Standard time = 11^57' ; index cor-
rection = +30"; declination at Greenwich Mean Noon = S 18 21' 08"; hourly
change = +39". 07; semidiameter = 16' 16".
Decl. at G. M. N. = - 18 21' 08" Double alt. = 59 17' 40"
39". 07 X 4^.95 +3' 13" I- C. __ 3
Decl. at n* s? m = - 18 17' 55" 2)59 18 10
29 39' 05"
Refraction = i 43
29 37 22
Semidiameter = 16 16
29 21 06
Parallax = ' +8
29 21 14
Declination 18 17 55
Colatitude = 47 39' 09"
Latitude = 42 20' 51"
67. By the Meridian Altitude of a Southern* Star.
The latitude may be found from the observed maximum alti-
tude of a star which culminates south of the zenith, by the
method of the preceding article, except that the parallax and
semidiameter corrections become zero, and that it is not necessary
to note the time of the observation, since the declination of the
star changes so slowly. In measuring the altitude the star's
image is bisected with the horizontal cross hair, and the maxi-
mum found by trial as when observing on the sun. For the
method of finding the time at which a star will pass the me-
ridian see Art. 72.
Observed meridian altitude of Serpentis = 51 45'; index correction = o;
declination of star = +4 05' n".
Observed altitude of Serpentis = 51 45' oo
Refraction correction = 46
Si 44 14
Declination of star = + 4 05 1 1
Colatitude = 47 39' 03"
Latitude = 42 20' 57"
* The observer is assumed to be in the northern hemisphere.
108 PRACTICAL ASTRONOMY
Constant errors in the measured altitudes may be eliminated
by combining the results obtained from circumpolar stars with
those from southern stars. An error which makes the latitude
too great in one case will make it too small by the same amount
in the other case.
68. Altitudes Near the Meridian.
If altitudes of the sun or a star are taken near the meridian they may be reduced
to the meridian altitude provided the latitude and the times are known approxi-
mately. To derive the formula for making the reduction take the fundamental
formula given in Equa. 
sin h = sin L sin D + cos L cos D cos P.
This may be transformed into
sin h = cos (L D) cos L cos D vers P, 
sin h = cos (L D) cos L cos D X 2 sin 2 
Transposing and denoting by h m the meridian altitude L D, the equation
sin h m = sin h + cos L cos D vers P, 
or sin h m = sin h + cos L cos D X 2 sin 2 
If the altitude h be measured and the corresponding time be noted, then the value
of P becomes known. If L is known approximately, then the second term may be
computed and h m , or L D, found through its sine. If the value of L derived
from the first computation does not agree closely with the assumed value, a second
computation should be made using the new value of L. When observations are
taken within a few minutes of the meridian (say 15) the computation may be
shortened by the use of the approximate formula
C" = 112.5 X P 2 X cos L cos D sec h sin i", 
in which C" is the correction in seconds of arc and P is the time from the meridian
expressed in seconds of time. (Log 112.5 X sin i" = 6.7367). If Pis expressed in
minutes the formula is
C" = i ".9635 X P 2 X cos L cos D sec h. 
This formula may be derived as follows: Transposing Equa. 
sin h m sin h = 2 cos L cos D sin 2 
2 cos | (h m + K) sin (h m - h) = 2 cos L cos D sin 2 
OBSERVATIONS FOR LATITUDE
Since h is nearly equal to h m , cos 5 (h m + h) may be put equal to cos h; placing
C = h m h, the equation becomes
sin 5 C = cos L cos .D sin 2 sec k. 
C and P are both small angles and may be put in place of their sines, hence
C = i P 2 X cos L cos D sec h. 
To reduce C to seconds of arc and P to seconds of time the left member must be
multiplied by sin i" and the right by (15 sin i") 2 , giving
C" = cos L cos D sec h X P 2 X 112.5 sin i". 
In using this formula it will be necessary to use an approximate value of L; a
second approximation may be made if necessary.
The method of " reduction to the meridian " given above should not be applied
when the object observed is far from the meridian.
Observed double altitude sun's lower limb Jan. 28, 1910.
Double Alt. Q
56 44' 40"
56 48' 47"
2) 56 40' i7"
28 24' 38'
Refr. = - 1 46
n* i5 m 25*
E. S. T.
n ft i6 m 19*
+ i 19
ii 57 21
s.d. = +16
Hour angle = 39 43*
P = 9 55' 45'
28 39 08
par. = +8
h = 28 39' 16'
log cos L = 9. 86763
log cos D = 9. 97745
log vers P = 8. 17546
nat sin h = . 47953
Assumed lat.. =
L. A. N.
L. M. T.
Red. for long. =
E. S. T.
Sun's decl. at
G. M. N. - -
39".o7 X 4*
12 OO OO
+ 13 03
12 13 03
ii 57 21
18 21' 08"
.3 = 2' 48"
nat sin h m = 49001
h m = 29 20' 29"
D = 18 18 20
Cor'd. decl. = 18 18' 20"
Co-lat. = 47 38' 49"
Lat. = 42 21' n"
A recomputation, using the corrected latitude, changes this result to 42 21' 04".
Observed altitude of y Ceti = 50 33'; index correction = i'; hour angle of
y Ceti derived from observed time = 3 m 14^.2; declination = -f- 2 50' 30".
log cos L = g. 8691
log cos D = 9. 9995
log sec h = o. 1967
log const = 6. 7367
2 log P = 4. 5765
= 23 ". 9
= i .o
Refraction correction = 0.8
True altitude = 50 31'. 2
Reduction to meridian = +0.6
= + 2
= 47" 4i'. 3
= 42 i8'. 7
The method of " ex-meridians altitudes," as it is sometimes called, may be used
when the meridian observation is lost or when it is desired to increase the accuracy
of the result by multiplying the number of observations.
69. Latitude by Altitude of Polaris when the Time is Known.
The altitude of Polaris varies slowly on account of its nearness to the pole,
hence, if the sidereal time is known, the latitude may be found accurately by an
altitude of this star taken at any time, because errors in the time have a rela-
tively small effect upon the result. Several altitudes should be taken in succession,
and the time noted at each pointing of the cross-hair on the star. For obser-
vations made with the surveyor's transit and covering only a few minutes' time
the mean of the altitudes may be taken as corresponding to the mean of the observed
times. If the instrument has a complete vertical circle, half of the observations
should be made with the instrument in the reversed position. The index correc-
tion should be determined in each case. In order to compute the latitude it is
necessary to know the hour angle of the star at the instant of observation. When
a common watch is used for taking the time the star's hour angle is found by Equa.
 and . The latitude is then found by the formula
L = h p cos P + | sin i' p 2 sin 2 P tan h 
(log \ sin i' = 6. 1627 10)
The derivation of the formula is rather complex and will not be given here. It
is obtained by expanding the correction to h in a series in ascending powers of
OBSERVATIONS FOR LATITUDE
p, the small terms being neglected. The sum of all terms after that containing
p 2 amounts to less than i" and these have therefore been omitted in Equa. .
In this equation p, the polar distance, is expressed in minutes of arc. Values of
the last term may be taken with sufficient accuracy from Table VI. The alge-
braic sign of the second term is deter-
mined by the sign of cos P; the third
term is always positive. In Fig. 54,
P is the pole, S the star, MS the
hour angle, and PDA the almucantar
through P or circle of equal altitudes.
It will be seen that the term p cos P
is approximately the distance from 5
to E, a point on the six-hour circle
PB; the distance desired is SD, the
angular distance of 5 above the al-
mucantar through P. The last term
in Equa.  is approximately equal
to DE, each term in the series giving
a closer approximation to the distance
Example. FIG. S4
Observed Altitudes of Polaris, Jan. 9, 1907.
6 h 49 26 s 43 28'. 5
5i 45 28.5
54 14 28 . o
56 45 28.0
Index correction = i'. o; p = 71'. 15; P is found from the observed watch times
to be 13 50 . 7.*
log p 1.8522 log constant =6.1627
log cos P =9. 9822 log p 2 = 3. 7044
log sin 2 P =8. 7578
log p cos P = i. 8394 log tan h =9. 9762
p cos P = 69'. 09
Observed alt. = 43 28'. 25
I.C. - i.
= + o'. 04
i st. and and terms
43 27'. 25
i . 01
43 26'. 24
i 09 .05
42 17'- 19
* If the error of the watch is known the sidereal time may be found by Equa. .
For method of finding the sidereal time by observation see Chap. XI. The
hour angle of the star is found by Equa. .
70. Precise Latitude Determinations. Talcott's Method.
The most precise method of determining latitude is that known as " Talcott's
Method," which requires the use of the zenith telescope. In making observations
the observer selects two stars, one north of the zenith and one south of it, the two
zenith distances differing by only a few minutes of angle, and the right ascensions
differing by about 5 or 10 minutes of time. For the best results the zenith dis-
tances should be small and nearly equal. If the first star culminates south of
the zenith the telescope is turned about its vertical axis until the stop indicates
that it is in the meridian and on the south side of the zenith. The telescope is
tipped until the sight line has an inclination to the vertical equal to the mean of
the two zenith distances.* It is clamped in this position and great care is taken
not to alter its inclination until the observations on both stars are completed.
When the star appears in the field the micrometer wire is set so as to bisect the
star's image; at the instant of culmination the setting of the wire is perfected and
the scale of the spirit level is read at the same time. The chronometer (regulated
to local sidereal time) should be read when the bisection is made, so that the read-
ing of the micrometer may be corrected if the star was not exactly on the meridian
at that instant. The micrometer screw is then read. The telescope is then turned
to the north side of the meridian, the inclination remaining unchanged, and a
similar observation made on the other star. When both sets of micrometer read-
ings and level readings have been obtained, the latitude is found by the formula
+ D n ) + } (m, - ) X R + i (/. + W + i (r s -r n ),
in which m s , m n are the micrometer readings, R the value of i division of the mi-
crometer, l s , l n the level corrections (positive when the north reading is the larger)
and r s , r n the refraction corrections. Another correction must be added in case
the observation is taken when the star is off the meridian.
In order to determine latitude by this method with the precision required in
geodetic operations, observations are made on several nights, and on each night
a large number of pairs of stars is observed. By this method a latitude may be
determined within about o". 05 which is equivalent to nearly 5 feet in distance
on the earth's surface.
Questions and Problems
1. Observed maximum altitude sun's lower limb, April 27, 1910, = 61 05'.
Index correction = + 30". The longitude is 4^ 44"* i8 s W. The sun's decl.
at G. A. N. = N 13 38' 22". 3; diff. for i h = +48". 07; the semidiameter =
15' 55". Compute the latitude.
2. The observed meridian altitude of 8 Crateris = 33 24'; index correction =
+ 30"; declination of star = 14 if 37". Compute the latitude.
3. Observed altitude of a Ceti at 3* o8 m 49* L. S. T. = 51 21'; I. C. = i'
* In order to compute these zenith distances it is necessary to know a rough
value of the latitude, say within i' or 2'. This may be found by an observation
with the zenith telescope using one of the preceding methods.
OBSERVATIONS FOR LATITUDE 113
the right ascension of a Ceti = 2 h 57"* 24*. 8; decimation = + 3 43' 22". Com-
pute the latitude.
4. Observed Altitude of Polaris, 41 41' 30"; chronometer time, g h 44 38*. 5
(Loc. Sid. Time); chronometer correction, 34*. The R. A. of Polaris is i* 25
42'; the declination is + 88 49' 29". Compute the latitude.
5. Show by a sketch the positions of the following three points; i. Polaris
at greatest elongation; 2. Polaris on the six-hour circle; 3. Polaris at the same
altitude as the pole.
6. What is the most favorable position of the sun for a latitude observation?
7. What is the most favorable position of Polaris for a latitude observation?
8. Draw a sketch showing why the sun's maximum altitude is not the same as
the meridian altitude.
OBSERVATIONS FOR DETERMINING THE TIME
71. Observation for Local Time.
Observations for determining the local time at any place at
any instant usually consist in finding the error of a timepiece
on the kind of time which it is supposed to keep. To find the
solar time it is necessary to determine the hour angle of the sun's
centre. To find the sidereal time the hour angle of the vernal
equinox must be measured. In some cases these quantities
cannot be measured directly, so it is often necessary to measure
other coordinates and to calculate the desired hour angle from
these measurements. The chronometer correction or watch
correction is the amount to be added algebraically to the read-
ing of the timepiece to give the true time at the instant. It is
positive when the chronometer is slow, negative when it is fast.
The rate is the amount the timepiece gains or loses per day;
it is positive when it is losing, negative when it is gaining.
72. Time by Transit of a Star.
The most direct and simple means of determining time is by
observing transits of stars across the meridian. If the line of
sight of a transit be placed so as to revolve in the plane of the
meridian, and the instant observed when some known star
passes the vertical cross hair, then the local sidereal time at this
instant is the same as the right ascension of the star given in
the Nautical Almanac for the date. The difference between
the observed chronometer time t and the right ascension a.
is the chronometer correction T,
or T = a-t. 
If the chronometer keeps mean solar time it is only necessary
to convert the true sidereal time a into mean solar time by
OBSERVATIONS FOR DETERMINING THE TIME 115
Equa. , and the difference between the observed and com-
puted times is the chronometer correction.
The transit should be set up and the vertical cross hair sighted
on a meridian mark previously established. If the instrument
is in adjustment the sight line will then swing in the plane of
the meridian. It is important that the horizontal axis should
be accurately levelled; the plate level which is parallel to this
axis should be adjusted and centred carefully, or else a striding
level should be used. Any errors in the adjustment will be
eliminated if the instrument is used in both the direct and re-
versed positions, provided the altitudes of the stars observed
in the two positions are equal. It is usually possible to select
stars whose altitudes are so nearly equal that the elimination
of errors will be nearly complete.
In order to find the star which is to be observed, its approxi-
mate altitude should be computed beforehand and set off on
the vertical arc. (See Equa. [i].) In making this computation
the refraction correction may be omitted, since it is not usually
necessary to know the altitude closer than 5 or 10 minutes.
It is also convenient to know beforehand the approximate time
at which the star will culminate, in order to be prepared for the
observation. If the approximate error of the watch is already
known, then the watch time of transit may be computed (Equa.
) and the appearance of the star in the field looked for a
little in advance of this time. If the data from the Nautical
Almanac are not at hand the computation may be made, with
sufficient accuracy for finding the star, by the following method :
Compute the sun's R. A. by multiplying 4 by the number of
days since March 22. Take the star's R. A. from any list of
stars or a star map. The star's R. A. minus the sun's R. A.
(Equa. ) will be the mean local time within perhaps 2"* or
3 TO . This may be reduced to Standard Time by the method
explained in Art. 35. In the surveyor's transit the field of view
is usually about i, so the star will be seen about 2 before it
reaches the vertical cross hair. Near culmination the star's
Il6 PRACTICAL ASTRONOMY
path is so nearly horizontal that it will appear to coincide with
the horizontal cross hair from one side of the field to the other.
When the star passes the vertical cross hair the time should be
noted as accurately as possible. A stop watch will sometimes
be found convenient in field obversations with the surveyor's
transit. When a chronometer is used the " eye and ear method "
is the best. (See Art. 58.) If it is desired to determine the
latitude from this same star, the observer has only to set
the horizontal cross hair on the star immediately after making
the time observation, and the reading of the vertical arc will
give the star's apparent altitide at culmination. (See Art. 67.)
The computation of the watch correction consists in finding
the true time at which the star should transit and comparing
it with the observed watch time. If a sidereal watch or chro-
nometer is used the star's right ascension is at once the local
sidereal time. If mean time is desired, the true sidereal time
must be converted into local mean solar time, or into Standard
Time, whichever is desired.
Observed transit of a Hydra 8 h 48"* 58.*. 5, Eastern time, in
longitude 5^ 20 west; date April 5, 1902. From the almanac,
the star's R. A. =g h 22 48 S .4, and the sun's R. A. at G. M. N. =
o h 51"* 24 S .6. To reduce this to the R. A. at local mean noon
take from Table III the correction for 5^ 20"* which is +52 S .6.
The corrected R. A. =o h 52 17*. 2. The local sidereal time,
which is g h 22 48 .4, is then reduced to Standard Time as
48 s .
Mean Local Time =8 29 07 .6
Red. to 75 merid. = 20 oo . o
Eastern Time =8 49 07.6
Watch time =8 48 58 . 5
Watch slow = 9 s . i
OBSERVATIONS FOR DETERMINING THE TIME 117
Transit observations for the determination of time can be
much more accurately made in low than in high latitudes.
Near the pole the conditions are very unfavorable.
73. Observations with Astronomical Transit.
The method just described is in principle the one in most common use for deter-
mining sidereal time with the large astronomical transit. Since the precision at-
tainable with the latter instrument is much greater than with the engineer's transit,
the method must be correspondingly more refined. The number of observations
on each star is increased by using a large number of vertical threads, commonly
eleven. These times of transit are recorded by electric signals on the chrono-
graph (see Art. 56, p. 93), and are scaled from the chronograph sheet to hundredths
of a second. In this class of work many errors which have been assumed to be
negligible in the preceding method are important and must be carefully determined
and allowed for. The instrument has to be set into the plane of the meridian by
means of repeated trials, and there is always a small remaining error in the azimuth
of the sight line. This error in azimuth a is measured by comparing the observed
times of rapidly moving (southern) stars and slowly moving (circumpolar) stars.
The correction to any observed time for the effect of azimuth error is
a cos h sec D. 
The inclination of the axis to the horizon b is measured with the spirit level and
the observed times are reduced to the meridian by adding the correction
b sin h sec D. 
The error in the sight line c is found by reversing the telescope in its supports
and comparing observations made in the two positions. The correction to any
c sec D. 
Corrections are also made for the effect of diurnal aberration and sometimes other
74. Selecting Stars for Transit Observations.
Before the observations are begun the observer should pre-
pare a list of stars suitable for transit observations. This
list should include the name or number of the star, its magni-
tude, the approximate time of culmination, and its meridian
altitude or its zenith distance. The right ascensions of consec-
utive stars in the list should differ by sufficient intervals to give
the observer time to make and record an observation and pre-
pare for the next one. The stars used for determining time
should be those which have a rapid diurnal motion, that is,
stars near the equator; slowly moving stars are not suitable
for time determinations. Very faint stars should not be selected
unless the telescope is of high power and good definition; those
smaller than the fifth magnitude are rather difficult to observe
with a small transit, especially as it is difficult to reduce the
amount of light used for illuminating the field of view. The
selection of stars will also be governed somewhat by a consider-
ation of the effect of the different instrumental errors. An in-
spection of Table B, p. 88, will show that for stars near the
zenith the azimuth error is zero, while the inclination error is
a maximum; for stars near the horizon the azimuth error is a
maximum and the inclination error is zero. If the azimuth of
the instrument is uncertain and the inclination can be accurately
determined, then stars having high altitudes should be preferred.
On the other hand, if the level parallel to the axis is not a sensi-
tive one and is in poor adjustment, and if the sight line can be
placed accurately in the meridian, which is usually the case
with a surveyor's transit, then low stars will give the more accu-
rate results. With the surveyor's transit the choice of stars is
somewhat limited, however, because it is not practicable to
sight the telescope at much greater altitudes than about 70
with the use of the prismatic eyepiece and 55 or 60 without
Following is a sample list of stars selected for observations
in a place whose latitude is 40 N., longitude 77 W., date May 5,
1910, hours between S h and 9^ Eastern time; the limiting alti-
tudes chosen are 10 and 65. The right ascension of the mean
sun for the date is 2 h 50. Adding this to 8 A - o8 m = 7* 52,
the local mean time, the resulting sidereal time is io h 42,
which is approximately the right ascension of a point on the
meridian at S h E. S. T. The limiting right ascensions are there-
fore io h 42 and n h 42. The co-latitude is 50, which gives,
for altitudes 10 and 65, the limiting declinations +15 and
40. In the table of mean places for 1910 the following
stars are given:
OBSERVATIONS FOR DETERMINING THE TIME
MEAN PLACES FOR 1910
IcA 44 W 2,2 s
+ 11 Ol'
5 J Chameleontis
TO 4.4 57
46 Leonis Minoris
TO 48 17
4- 2.4 42
Groombridge 1 706
IO C2 47
+ 78 I?
a Ursa Majoris
10 58 ii
+ 62 14
10 so s8
p 3 Leonis
II O2 IQ
\f/ Urstz Majoris
II 04 2. 7
+ 44 <O
2 . 7
1 1 OQ I Q
+ 21 O I
v Ursce Majoris
II 1 3 2.7
+ ?"? ^?
II 14 !JO
II 21 IO
+ 7 21
II 26 04
+ 60 sO
v Hydra ...
II 28 74
' v Leonis
X Ursa Majoris
ii 41 18
TT 4/1 28
-l-i c os
From this list there are found seven stars whose declinations
and right ascensions fall within or very close to the required
limits. In the following list the times of transit and the alti-
tudes have been computed roughly but with sufficient accuracy
to identify the stars.
OBSERVING LIST FOR TRANSIT OBSERVATIONS
8 h oo m
c . i
75- Time by Transit of the Sun.
The apparent solar time may be directly determined by
observing the watch times when the west and east limbs of the
120 PRACTICAL ASTRONOMY
sun cross the meridian. The mean of the two readings is the
watch time for the instant of Local Apparent Noon or i2 h M..
apparent time. This apparent time is to be converted into
mean time and then into Standard Time. If only one limb
of the sun can be observed the time of transit of the centre may
be found by adding or subtracting the " time of semidiameter
passing the meridian," which is given in the Nautical Almanac.
Observed transit of sun on Jan. 28, 1910, longitude 4* 44* i8 s W. Time of
transit of W. limb = n h 54 53*; E. limb = n h 57"* n s ; mean of two limbs =
ii* 56 m 02*. o.
L. M. T. = i2 h oo m oo s Equa. of T. at G. A. N. =13"* oo s . 71
Equa. T. = +13 03 . o o s . 485 X 4 A . 74 = 2 .30
L. M. T. =12 13 03.0 Cor'd. Equa. T. = 13 03*. 01
Red. to 75 = 15 42 .o
E.S.T. =11 57 21.0
Watch time = 1 1 56 02.0
Watch slow = i m i9 s . o
. 76. Time by an Altitude of the Sun.
The apparent solar time may be determined by measuring
the altitude of the sun when it is not near the meridian, and
then solving the PZS triangle for the angle at the pole, which is
the hour angle of the sun east or west of the meridian. The
west hour angle of the sun is the local apparent time. The
observation is made by measuring several altitudes in quick
succession and noting the corresponding instants of time. The
mean of the observed altitudes is assumed to correspond to the
mean of the observed times, that is, the curvature of the path
of the sun is neglected. The error caused by neglecting the
correction for curvature is very small provided the sun is not
near the meridian and the series of observations extends over
but a few minutes' time, say io m . The measurement of alti-
tude must of course be made to the upper or the lower limb
and a correction applied for the semidiameter. The observa-
tions may be made in two sets, half the altitudes being taken
on the upper limb and half on the lower limb, in which case no
semidiameter correction is required. The telescope should be
OBSERVATIONS FOR DETERMINING THE TIME 121
reversed between the two sets if the instrument has a complete
vertical circle. The mean of the altitudes must be corrected
for index error, refraction, and parallax, and for semidiameter
if but one limb is observed. The declination must be corrected
by adding to the declination at G. M. N. the hourly change mul-
tiplied by the number of hours since G. M. N. It is necessary
for this purpose to know the approximate Greenwich Mean
Time. If the watch used is keeping Standard Time the G. M. T.
is found at once. (Art. 35.) If the watch is not more than
2 m or ^ m j n error the effect on the computed declination will be
negligible for observations made with small instruments. If the
longitude is known the declination may be corrected by first
computing an approximate value of the local time and adding
this to the longitude, obtaining the approximate G. M. T.
With this approximate G. M. T. the declination may be cor-
rected and the whole computation repeated. It will seldom be
necessary to make a third computation. In order to compute
the hour angle the latitude of the place must be known. The
hour angle of the sun's centre P is then found by means of one
of the formulae of Art. 19.* When the value of P is found it
is converted into hours, minutes and seconds, and if the sun is
east of the meridian it is subtracted from 1 2 h to obtain the local
(civil) apparent time; if astronomical time is desired it should
be subtracted from 24^. This apparent time is then converted
into mean time by adding or subtracting the equation of time.
* If tables of log versed sines, in addition to the usual tables, are available,
then the following formula will be found convenient:
_ cos (L D) sin k
vers P = , ,. 83
cos L cos D
In case P is greater than 90 the formula below may be substituted:
vers p. = sinA + co S (L + Z)) )
cos L cos D
where P' = 180 - P.
The sum or difference in the numerator must be computed with natural func-
tions and the remainder of the computation performed by logarithms.
The equation of time must be corrected for the time elapsed
since G. M. N. The resulting mean time is to be converted
into Standard Time, to which the watch is regulated. The
difference between the computed result and the mean of the
observed watch readings is the watch correction.
The most favorable conditions for an accurate determination
of time by this method are when the sun is on the prime vertical
and when the observer is at the equator. When the sun is
east or west it is rising or falling at its most rapid rate and an
error of i' in the altitude produces less error in the calculated
hour angle than does i' error when the sun is near the meridian.
The nearer the observer is to the equator the greater is the in-
clination of the sun's path to the horizon, and consequently the
greater its rise or fall per second of time. If the observer were
at the equator and the declination zero, the sun would rise or
fall i' in 4 s of time. In the example given below the rise is
i' in about 8 s of time. When the observer is near the pole
the method is practically useless. Observations on the sun
when it is very close to the horizon should be avoided, however,
even when the sun is near the prime vertical, because the errors
in the tabulated refraction correction due to variations in the
temperature and pressure of the air are likely to be large. Ob-
servations should not be made when the altitude is less than
about 10 if this can readily be avoided.
Observation of Sun's Altitude for Time, Nov. 28, 1905. Lat. 42 21' N. Long.
71 04. 5' W.
j IS? 55'
1 6 08'
Refraction and parallax =
h = 15 22'. 7
Watch (Eastern Time)
& h 3Q m 42 s A.M.
8 42 19
Mean = 8 h 43 47*. 2 A.M.
G. M. T. = i h 43 47. 2 (approx.)
OBSERVATIONS FOR DETERMINING THE TIME 123
L 42 21'. o sec o. 13133 Decl. at G. M. N. = 21 14' 54"
h = 15 22.7 / -26".8iXA 73 = -46
P = in 15.7 esc 0.03061
Corrected decl. = 21 15' 40"
25 = 178 59'. 4 p = 111 15' 40"
s = 84 29'. 7 cos 8. 98196
s h = 69 07'. o sin 9. 97049
Eq. t. = i2 m 04 s . 29
2)9.11439 .846X1.73 = I-46
log sin P = 9. 55719 i2 m 02 s . 83
i P = 21 08' 45"
P = 42 17' 30"
= 2 h 49 io s . o
k, A , T - ' :r 5:i
j"= j. //,+. -**>)
Zl, t r *
M.L.T. =8* 58 47 s - 2
15 42 .o
Eastern time = 8 h 43"* 05" . 2
Watch time = 8 43 47 . 2
Watch fast 42 s . o
77. Time by the Altitude of a Star.
The method of the preceding article may be applied equally
well to an observation on a star. In this case the parallax and
semidiameter corrections are zero. If the star is west of the meri-
dian the computed hour angle is the star's true hour angle;
if the star is east of the meridian the computed hour angle must
be subtracted from 24^. The sidereal time is then found by
adding the right ascension of the star to its hour angle. If
mean time is desired the sidereal time thus found is to be con-
verted into mean solar time by Art. 34. Since it is easy to select
stars in almost any position it is desirable to eliminate errors in
the measured altitudes by taking two observations, one on a
star which is nearly due east, the other on one about due west.
The mean of these two results will be nearly free from instru-
mental errors, and also from errors in the assumed value of the
observer's latitude. If a planet is used it will be necessary to
know the G. M. T. with sufficient accuracy for correcting the
right ascension and declination.
Observed altitude of
Refraction correction =
L = 42 iS'.o
h = 44 53-o
p = 66 41 . 4
Jupiter (east), Jan. 9, 1907. Lat. = 42 18'. o; Long. =
Eastern time = 7* 32 02 s
- i Decl. at G. M. N. = + 23 18' 22". o
i Hourly change = + i". oo
G. M. T. = i2 h 32 02 s (approx.)
44 53'- o* 12^.53 X i ".oo = +12". 5
Corrected decl. = + 23 18' 34". 5
p = 66 41' 25". 5
* 17 s - 3
- i 8 - 395
-17 s . 5
1 59 s - 8
03 . 2
2o* 41 W
6* i8 m
12 s .
59 s -
2 S = 152 112 . 4
S 76 56 . 2
R. A. at G. M. N. = 6* 19'
1 2 h . 53 X - i. 395 =
Corrected R. A. = 6* 18'
Sid. Time = 27* oo" 1 12 s . 2
The local sidereal time is therefore 3^ oo w 1 2 s . 2 when the watch reading is 7 * 3 2 TO 02 s .
The error of the watch may be found by reducing the sidereal time to Eastern Time.
78. Time by Transit of Star over Vertical Circle through Polaris.|
In making observations by this method the line of sight of the telescope is set
in the vertical plane through Polaris at any (observed) instant of time, and the
time of transit of some southern star across this plane is observed immediately
afterward; the correction for reducing the star's right ascension to the true sidereal
time of the observation is then computed and added to the right ascension. The
advantages of the method are that the direction of the meridian does not have to
be established before time observations can be begun, and that the interval which
must elapse between the two observed times is so small that errors due to the
instability of the instrument are reduced to a minimum.
The method of making the observation is as follows: Set up the instrument and
level carefully; sight the vertical cross hair on Polaris (and clamp) and note and
record the watch reading; then revolve the telescope about the horizontal axis,
* Parallax is negligible for this planet, as it is only about 2".
f This method is given by Mr. George O. James in the Jour. Assoc. Eng. Soc.
Vol. XXXVII, No. 2. In a later paper (Popular Astronomy No. 172) Mr.
James gives the formula
P = p sec L sin (L - D) sec (D - c) sin (P P),
in which c is the correction from Table IV in the Nautical Almanac. This for-
mula is preferable to that given in the text, provided the latitude is known, since
it is not necessary to make a second approximation. A discussion of the method
used with large instruments is given by Professor Frederick H. Sears in Bul-
letin No. 5, Laws Observatory, University of Missouri.
OBSERVATIONS FOR DETERMINING THE TIME
being careful not to disturb its azimuth; set off on the vertical arc the altitude
of some southern star (called the time-star) which will transit about 4 or $ m
later; note the instant when this star passes the vertical cross hair. It will be of
assistance in making the calculations if the altitude of each star is measured
immediately after the time has been observed. The altitude of the time-star
at the instant of observation will be so nearly equal to its meridian altitude that
no special computation is necessary beyond what is required for ordinary transit
observations. If the times of meridian transit are calculated beforehand the
actual times of transit may be estimated with sufficient accuracy by noting the
position of Polaris with respect to the meridian. If Polaris is near its elongation
then the azimuth of the sight
line will be a maximum. In
latitude 40 the azimuth of
Polaris for 1910 is about i 32';
a star on the equator would
then pass the vertical cross hair
nearly 4 later than the com-
puted time if Polaris is at
eastern elongation (see Table
B). If Polaris is near western
elongation the star will transit
earlier by this amount. In
order to eliminate errors in the
adjustment of the instrument,
observations should be made
in the erect and inverted posi-
tions of the telescope and the
two results combined. A new
setting should be made on
Polaris just before each obser-
vation on a time-star.
In order to deduce an expression for the difference in time between the meridian
transit and the observed transit let R and R be the right ascensions of the stars,
5 and So the sidereal times of transit over the cross hair, P and P the hour angles
of the stars, the subscripts referring to Polaris. Then by Equa. ,
P =5 - R
and Po = So- RO]
subtracting, P - P = (R - ) - (5 - So). 
The quantity 5 S is the observed interval of time between the two observa-
tions expressed in sidereal units. If an ordinary watch is used the interval must
be reduced to sidereal units (Table III). Equa.  may then be written
Po - P = (R - Ro) - (T - To) - C, 
where T and T are the actual watch readings and C is the correction to reduce
this interval to sidereal time^-
In Fig. 55 let P be the position of Polaris; P, the celestial pole; Z, the zenith;
and S, the time-star. Also let Z and Zo represent the azimuths of the two stars;
p and PO, their polar distances; z and z , their zenith distances; and h and ha,
their altitudes. In the triangle PPoZ,
and in the triangle PZS
sin P _ sin Zo
sin zo sin /><>'
sin P sin Z
Since the azimuth of the sight line has not changed,
smnl = 1 80 + sm
or Z = 180 + Z
and sin Z = sin Z .
From Equa.  and , solving for sin P, there results
sin P = sin p sin P sec Z) cos h sec
in which D is the declination of the time-star 5. Since the angles P and p Q are
small they may be substituted for their sines, giving
P = po sin PO sec D cos h sec &o. 
In this equation the value of PO is unknown, and unless the local sidereal time is
already known with accuracy it is necessary to determine PO by a series of approxi-
mations. A rough value of PO ( = P'o) may be found from the equation
P'o = (R- *) - (T - r ) - C. 
Using this value of P' in Equa.  the result is P', an approximate value of P.
A corrected value of PO is then obtained by the equation
Po = P'oV P'- 
With this new value of PO a new value of P may be computed. If the second
value of P differs much (say 5') from the first value it will be necessary to make
another computation of PO. It is usually possible to make a rough estimate of
PO from the known value of the local time. The watch time of the observation
on Polaris may be converted into local sidereal time and the hour angle P found
by Equa.  and . When a series of observations is made the hour angle
of Polaris at all observations after the first may be closely estimated by adding
to the value of PO at the first observation the time elapsed since the first obser-
vation on Polaris. If the altitudes of the stars have not been measured it is
usually accurate enough to take h = 90 L D (the meridian zenith distance)
for the time-star, and for Polaris ho = L + po cos P , or better still h = L c,
where c is the quantity given in Table IV at the end of the Nautical Almanac.
The final value of P, the hour angle of the time-star at the instant it was observed,
is the correction to be added to the right ascension of the time-star to obtain the
local sidereal time of the observation on this star. This sidereal time may then be
reduced to mean time or to standard time and the watch correction obtained.
OBSERVATIONS FOR DETERMINING THE TIME 127
The above method is applicable to transit observations made with a small
instrument. For the large astronomical transit a more refined method of making
the reductions must be used.
Observation of o Virginis over Vertical Circle through Polaris; Lat.,42 21' N.,
Long., 4^ 44"* 18*. 3 W.; Date, May 8, 1906.
Observed time on Polaris = 8 h 35"* 58 s
Observed transit of o Virginis = 8 39 43
Diff. = 3 45*
R = i2 h oo m 26 s . 3
Ro= i 24 35 .4 L = 42 21'
D= + 9 15
R R = io h 35 50*. 9
T - T = 3 45 .o L-D= 33 06'
C = .6
#0 = ?i'. 85
P'o = IOJ 3 2 OSM - log #o=I. 8564
, Z 5 8 OI r -3 log sec D = 0.0057-
P -19 -8 log sin (L- D) = 9. 7373
D , , D , o , log sec (L - c) = o. 1238
P'o+P = i57 41-5 log sin P' = 9.5732
log P' = i. 2964 w
P'= -19'. 79
The log sin of (P' + P') = 9-5793; substituting this for log sin P' , the log P'
is increased 61 units in the fourth place, giving 20'. 07 for P. Converting this
into time it is 8o s . 3 or i m 20". 3, the desired correction. The true sidereal
time may now be found by subtracting i m 2o s . 3 from the right ascension of
o Virginis. The complete computation of the watch correction is as follows:
R = T 2 h oo m 26 s . 3
P = i 20 . 3
5 = n h 59 TO o6 s .o
R s = 3 p2 23 .6
&h 5 6" 428.4
C = i 27 .9
M. L. T.= 8 h 55 14*. 5
15 4i -7
Eastern time = 8^ 39 32*. 8
Watch time =8 39 43
Watch fast = io s . 2
79. Time by Equal Altitudes of a Star.
If the altitude of a star is observed when it is east of the meridian at a certain
altitude, and the same altitude of the same star again observed when the star is
west of the meridian, then the mean of the two observed times is the watch reading
for the instant of transit of the star. It is not necessary to know the actual
value of the altitude employed, but it is essential that the two altitudes should be
equal. The disadvantage of the method is that the interval between the two
observations is inconveniently long.
80. Time by Two Stars at Equal Altitudes.
In this method the sidereal time is determined by observing when two stars
have equal altitudes, one star being east of the meridian and the other west. If
the two stars have the same declination then the mean of the two right ascensions
is the sidereal time at the instant the two stars have the same altitude. As it is
not practicable to find pairs of stars having exactly the same declination it is neces-
sary to choose pairs whose declinations differ as little as possible and to introduce
a correction for the effect of this difference upon the sidereal time. It is not
possible to observe both stars directly with a transit at the instant when their
altitudes are equal; it is necessary, therefore, to first observe one star at a certain
altitude and to note the time, and then to observe the other star at the same alti-
tude and again note the time. The advantage of this method is that the actual
value of the altitude is not used in the computations; any errors in the altitude
due either to lack of adjustment of the transit or to abnormal refraction are there-
fore eliminated from the result, provided the two altitudes are made equal. In
preparing to make the observations it is well to compute beforehand the approx-
imate time of equal altitudes and to observe the first star two or three minutes
before the computed time. In this way the interval between the observations
may be kept conveniently small. It is immaterial whether the east star is observed
first or the west star first, provided the proper change is made in the computation.
If one star is faint it is well to observe the bright one first; the faint star may then
be more easily found by knowing the time at which it should pass the horizontal
cross hair. The interval by which the second observation follows the time of
equal altitudes is nearly the same as the interval between the first observation
and the time of equal altitudes. It is evident that in the application of this
method the observer must be able to identify the stars he is to observe. A star
map is of great assistance in making these observations.
The observation is made by setting the horizontal cross hair a little above the
easterly star 2 or 3"* before the time of equal altitudes, and noting the instant
when the star passes the horizontal cross hair. Before the star crosses the hair
the clamp to the horizontal axis should be set firmly, and the plate bubble which
is perpendicular to the horizontal axis should be centred. When the first obser-
vation has been made and recorded the telescope is then turned toward the westerly
star, care being taken not to alter the inclination of the telescope, and the time
when the star passes the horizontal cross hair is observed and recorded. It is
well to note the altitude, but this is not ordinarily used in making the reduction.
If the time of equal altitudes is not known, then both stars should be bright ones
that are easily found in the telescope. The observer may measure an approxi-
mate altitude of first one and then the other, until they are at so nearly the same
altitude that both can be brought into the field without changing the inclination
of the telescope. The altitude of the east star may then be observed at once and
OBSERVATIONS FOR DETERMINING THE TIME
the observation on the west star will follow by only a few minutes. If it is desired
to observe the west star first, it must be observed at an altitude which is greater
than when the west star is observed first. In this case the cross hair is set a little
below the star.
In Fig. 56 let nesw represent the horizon, Z the zenith, P the pole, S e the easterly
star, and S w the westerly star.
Let P e and P w be the hour
angle of S e and S w , and let
HS e Sw be an almucantar, or
circle of equal altitudes.
From Equa. , for the
two stars S e and S w , the
sidereal time is
5 = R w + Pw
S = R e - Pe*
Taking the mean value of 5,
Rw~T~Re . Pw Pe r i
s= - +-., 
from which it is seen that
the true sidereal time equals
the mean right ascension
corrected by half the differ-
ence in the hour angles. To
derive the equation for cor-
recting the mean right ascension so as to obtain the true sidereal time let the
sin h = sin D sin L + cos D cos L cos P 
be differentiated regarding D and P as the only variables, then there results
o = sin L cos D cos D cos L sin P -=- cos L cos P sin D,
from which may be obtained
tan L tan D
sin P tan P
If the difference in the declination is small, dD may be replaced by (D w D e ),
in which case dP will be the resulting change in the hour angle, or 5 (P w P e ).
The equation for the sidereal time then becomes
R w + Re . Dw - D e Ran L _ tan D~\
sinP tanP '
in which (D w D e ) must be expressed in seconds of time. D may be taken
as the mean of D e and D w . The value of P would be the mean of P e and P w if
P e is here taken as the actual value of the hour angle east of the meridian.
the two stars were observed at the same instant, but since there is an appreciable
interval between the two times P must be found by
n ~n T* *T*
p = R^_R + T^T. [9fi]
If the west star is observed first, then the last term becomes a negative quantity.
Strictly speaking this last term should be converted into sidereal units, but the
effect upon the result is usually very small. In regard to the sign of the correction
to the mean right ascension it should be observed that if the west star has the
greater declination the time of equal altitudes is later than that indicated by the
mean right ascension. In selecting stars for the observation the members of a
pair should differ in right ascension by 6 to 8 hours, or more, according to the
declinations. Stars above, the equator should have a longer interval between
them than those below the equator. On account of the approximations made in
deriving the formula the declinations should differ as little as possible. If the
declinations do not differ by more than about 5, however, the result will usually
be close enough for observations made with the engineer's transit. From the
extensive star list now given in the Nautical Almanac it is not difficult to select
a sufficient number of pairs at any time for making an accurate determination
of the local time. Following is a short list taken from the American Ephemeris
and arranged for making an observation on April 30, 1912.
LIST FOR OBSERVING BY EQUAL ALTITUDES
Lat., 42 21' N. Long., 4* 44 18* W. Date, Apr. 30, 1912.
of equal alti-
of equal alti-
a Corona Borealis
2 . T.
ft Tauri .
7 h 3 gm
7 5 8
ft Herculis . . ...
rj Geminorum. ... ...
2 . 7
a Cnnis Minoris
2 . I
OBSERVATIONS FOR DETERMINING THE TIME 131
Following is an example of an observation for time by the method of equal alti-
Lat, 42 21' N. Long., 4* 44 i8 s W. Date, Apr. 14, 1905.
Star. Rt. Asc. Decl. Watch.
a Ceti (E) 2* S7" 22 s . i + 3 43' 69". i 5* i8 oo
8 Aquila (W) 19 20 43-6 +2 55 44 .o 5 22 13
Mean 23* 09 02 s . 8 +3 19' 56". 6 $ h 20 06*. 5
Diff. 7 36 38 . 5 2)-o 48' 25".! 04 13
4 *3 7 D w -D f ,
- - e = 24 12 .6
2) 7* 4Q 52*. i 2 _ _ Q6 . 8 .
P= 3 h 50^26^ 9& ' 84
= 57 36' 31". 5
Mean R. A. = 23^ og m 02*. 8
Corr. = 01 41.0
Sid. Time = 23^ o?" 1 21*. 8 r> n
T> i ^-'u> J^e o^/\ o/-/\
^s = 17 30 43 . 2 log - = 1.9861 (n) 1.9861 (n)
rfc ?6 W i8 s 6 lo S tan L = 9- 9598 log tan Z> = 8. 7650
55 2 csc P ~ - 735 log cot P ~ 9-
MLT = c* 4V 4. =2. 0194 (n) 0.5535 (n)
15 42 o -f 6 6 -3.6
Eastern time = 5^ 2o m oi s . 4
Watch time =5 20 06 . 5 Corr -
Watch fast = 5 s . i
8 1 . Formula  may be made practically exact by means of the following device.
Applying Equa.  to each star separately and subtracting one result from the
other we obtain the equation*
tan L tan AZ? tan D tan AZ? . tan D tan AZ?
sin AP = r~ s vers AP, k)?!
sin P tan P tan P
where AZ? is half the difference in the declinations and AP is the correction to
the mean right ascension. If sin AP and tan AZ? are replaced by their arcs
and the third term dropped, this reduces to Equa. , except that AZ? and AP
are finite differences instead of infinitesimals. In order to compensate for the
errors thus produced let AZ? be increased by a quantity equal to the difference
between the arc and the tangent (Table C); and let a correction be added to the
sum of the first two terms to allow for the difference between the arc and sine of
AP (Table C). With the approximate value of AP thus obtained the third
* Chauvenet, Spherical and Practical Astronomy, Vql. I, p. 199.
term of the series may be taken from Table D. By this means the precision of
the computed result may be increased, and the limits of AZ) may therefore be
extended without increasing the errors arising from the approximations.
TABLE C. CORRECTIONS TO BE ADDED TO AZ> AND AP
(Equa. , Art. 81)
Arc or sine.
Arc or sine.
TABLE D. CORRECTION TO BE ADDED TO AP *
(Equa. , Art. 81)
AP (in seconds of time).
* The algebraic sign of this term is always opposite to that of the second term.
OBSERVATIONS FOR DETERMINING THE TIME 133
Compute the time of equal altitudes of a Bootis and i Geminorum on Jan. i,
1912, in latitude 42 21'. R. A. a Bootis = 14^ n m 37*. 98; decl. = + 19
38' 15". 2. R. A. i Geminorum = ? h 20 i6 s . 85; decl. = + 27 58' 30". 8.
I4 h Iim 37 s. 9 8 27 58' 30". 8
7 20 16 .85 19 38 15 . 2
2) 6 h 51 2i s . 13 2)8 20' 15". 6
3^ 25"* 40*'. 56 AZ> = 4 10' 07". 8
P = 51 25' 08". 4 = 1000 s . 52
Corr., Table C = i . 77
AZ? =1002*. 29
log AZ> = 3.000993 log AD = 3.00099
log tan L =9. 959769 log tan D = 9. 64462
log esc P =o. 106945 log cot P =9. 901. 87
1168'. 71 352-76
352 . 76
8 1 5 s - 95
Corr., Table C = + .48
Corr., Table D = + . 63
AP = + 8i7.o6
= + 13"* 37 s - 06
Mean R. A. = 10 45 57 . 42
Sid. Time of Equal Alt. = io h 59 34*. 48
For refined observations the inclination of the vertical axis should be measured
with a spirit level and a correction applied to the observed time. With the engi-
neer's transit the only practicable way of doing this is by means of the plate-level
which is parallel to the plane of motion of the telescope. If both ends of this
level are read at each observation, denoting the reading of the object end and E
the eye end of the bubble, then the change in the inclination is expressed by
where d is the angular value of one scale division in seconds of arc. The correction
to the mean watch reading is
_ _ i _ _ _ i _
30 sin 5 cos D 30 cos L sin Z
in which 5 may be taken from the Azimuth* tables or Z may be found from the
measured horizontal angle between the stars. If the west star is observed at a
higher altitude than the east star (bubble nearer objective), the correction must
be added to the mean watch reading. If it is applied to the mean of the right
ascensions the algebraic sign must be reversed.
* See Arts. 82 and 109 for the method of using these tables.
82. The correction to the mean right ascension of the two stars may be con-
veniently found by the following method, provided the calculation of the paral-
lactic angle, 5 in the PZS triangle, can be avoided by the use of tables. Publica-
tion No. 1 20 of the U. S. Hydrographic Office gives value of the azimuth angle
for every whole degree of latitude and declination and for every io m of hour angle.
The parallactic angle may be obtained from these tables (by interpolation) by
interchanging the latitude and the declination, that is, by looking up the declin-
ation at the head of the page and the latitude in the line marked " Declination."
For latitudes under 23 it will be necessary to use Publication No. 71.
In taking out the angle the table should be entered with the next less whole
degree of latitude and of declination and the next less io m of hour angle, and the
corresponding tabular angle written down; the proportional parts for minutes
of latitude, of declination, and of hour angle are then taken out and added alge-
braically to the first angle. The result may be made more accurate by working
from the nearest tabular numbers instead of the next less. The instructions given
in Pub. 1 20 for taking out the angle when the latitude and declination are of
opposite sign should be modified as follows. Enter the table with the supplement
of the hour angle, the latitude and declination being interchanged as before, and
the tabular angle is the value of 5 sought.
Suppose that two stars have equal declinations and that at a certain instant
their altitudes are equal, A being east of the meridian and B west of the meridian.
If the declination of B is increased so that the star occupies the position C, then
the star must increase its hour angle by a certain amount x in order to be again
on the almucantar through B. Half of the angle x is the desired correction.
In Fig. 57 BC is the increase in declination; BD is the almucantar through
A, B and D; and CD is the arc of the parallel of declination through which the
star must move in order to reach BD. The arcs BD and CD are not arcs of
grfet circles, and the triangle BCD is not strictly a spherical triangle, but it may
OBSERVATIONS FOR DETERMINING THE TIME 135
be shown that the error is usually negligible in observations made with the engi-
neer's transit if BCD is computed as a spherical triangle or even as a plane triangle.
The angle ZBP is the angle S and DEC is 90 - 5. The length of the arc CD
is then BC cot S, or (D w D e ) cot S. The angle at P is the same as the arc
CD' and equals CD sec D. If (D w D e ) is expressed in minutes of arc and the
correction is to be in seconds of time, then, remembering that the correction is
half the angle x,
Correction = 2 (D w D e ) cot S sec D. 
D should be taken as the mean of the two declinations, and the hour angle, used
in finding S, is half the difference in right ascension corrected for half the watch
The trigonometric formula for determining the correction for equal altitudes is
tan = sin cot J (Si + S 2 ) sec j (D l + D 2 ). 
By substituting arcs for the sine and tangent this reduces to the equation given
above, except that the mean of Si and S 2 is not exactly the same as the value of S
obtained by using the mean of the hour angles.
The example on p. 131 worked by this method is as follows. From the azimuth
tables, using a declination of 42, latitude 3, and hour angle 3* SO TO , the approxi-
mate value of S is 44 05'. Then from the tabular differences,
Correction for 21' decl. = 22'
Correction for 20' lat. = + 07
Correction for 26* h. a. = + 02
The corrected value of S is therefore 43 52'
2 (D w - D e ) = - 96'. 84 log = i. 9861 (n)
log cot S = o. 0172
log sec D = o. 0007
log corr. = 2. 0040 (n)
log corr. = ioo s . 9
This solution is sufficiently accurate for observations made with the engineer's
transit, provided the difference in the declinations of the two stars is not greater
than about 5 and the other conditions are favorable. For larger instruments
and for refined work this formula is not sufficiently exact.
The equal-altitude method, like all of the preceding methods, gives more precise
results in low than in high latitudes.
83. Rating a Watch by Transit of a Star over a Range.
If the time of transit of a fixed* star across some well-defined
range can be observed, the rate of a watch may be quite accu-
rately determined without knowing its actual error. The
disappearance of the star behind a building or other object
* A planet should not be used for this observation.
when the eye is placed at some definite point will serve the pur-
pose. The star will pass the range at the same instant of sidereal
time every day. If the watch keeps sidereal time, then its
reading should be the same each day at the time of the star's
transit over the range. If the watch keeps mean time it will
lose 3 m 55 s .Qi per sidereal day, so that the readings on successive
days will be less by this amount. If, then, the passage of the
star be observed on a certain night, the time of transit on any
subsequent night is computed by multiplying 3 55 s -9i by the
number of days intervening and subtracting this correction
from the observed time. The difference between the observed
and computed times divided by the number of days is the daily
gain or loss. After a few weeks the star will cross the range in
daylight, and it will be necessary before this occurs to transfer
to another star which transits later in the same evening. In
this way the observations may be carried on indefinitely.
84. Time Service.
The Standard Time used for general purposes in this country
is determined by observations at Washington and is sent out
to all parts of the country east of the Rocky Mountains by
means of electric signals transmitted over the lines of the tele-
graph companies. For the territory west of the Rocky Moun-
tains the time is determined at the Mare Island Navy Yard
and distributed by telegraphic signals. The error of the sidereal
clock of the observatory is determined at frequent intervals
by observing star transits. The sidereal clock is then compared
with a mean-time clock, by means of a chronograph, and the
error of this clock on mean time is computed. The mean-time
clock is then compared with another mean-time clock especially
designed for sending the automatic signals. When the error
of this sending clock is found it is " set " (to Eastern Standard
Time) by accelerating or retarding the motion of the pendulum
until the error is reduced to a negligible quantity. The series
of signals sent out each day begins at 11^55 ""A.M., Eastern
time, and continues for five minutes. The clock mechanism
OBSERVATIONS FOR DETERMINING THE TIME 137
is arranged to break the circuit at the end of each second;
this makes a click on every telegraph sounder on the line, or
a notch on the sheet of a chronograph placed in the circuit.
The end of each minute is shown by the omission of the 55th
to 59th seconds inclusive, except for the noon signal, which
is preceded by a ten-second interval. During this ten-second
interval the local circuits controlling the time-balls,* which
are dropped by this same signal, are thrown into the main
circuit. The signals sent out in this way are seldom in error
by an amount greater than one or two tenths of a second. The
break in the circuit which occurs at the instant of noon, Eastern
time, drops all the time-balls, corrects the clocks placed in the
circuit, and gives a click on every telegraph sounder on the line.
In many seaports the wireless telegraph lines are also thrown
into the circuit and the signal thus made available at sea.
Questions and Problems
1. Compute the approximate Eastern time of transit of Regulus over the me-
ridian 71 04/.5 West of Greenwich on March 21, 1908. The R. A. of Regulus
is io h 03 m 29'.!; R s at G. M. N. = 23^ 54 23*.99.
2. Compute the error of the watch from the data given in prob. 6, p. 169.
3. Observed time of transit of 5 Capricorni over the vertical circle through
Polaris, Oct. 26, 1906. Latitude = 42 iS'.s; longitude = 4 h 45 07*. Ob-
served watch time of transit of Polaris = 7 h io m 20 s ; of d Capricorni = 7^ 13 28 s ,
Eastern Time. Declination of Polaris = + 88 48' 31 ".3; right ascension =
i h 26 37 s -9- Declination of d Capricorni = 16 3.3' 02". 8; right ascension =
2I h ^w 538.2. The right ascension of the Mean Sun at Local Mean
Noon = i4 h i6 m 34 S .6. Compute the error of the watch on Eastern Time.
4. Time observation on May 3, 1907, in latitude 42 2i'.o, longitude 4^ 44
i8 s .o. Observed transit of Polaris = *] h i6 m 17 '.o; of M Hydra = -j h i& m 5o s -5.
Decl. of Polaris = + 88 48' 28"-3; R. A. = i h 24 so s .2. Decl. of M Hydra =
- i62i's3".2;R. A.= 10^21 36*.!. R. A. of Mean Sun at G. M. N. = 2^40
563.63. Find the error of the watch.
5. Observation for time by equal altitudes, Dec. 18, 1904.
R. A. Decl. Watch.
a Tauri (E) 4 h 3 m 2 9 s -i + l6 18' S9"-9 j h 34 56*
a Pegasi (W) 22 59 61 .12 + 14 41 43 .7 7 39 45
Lat. = 42 28'.o; long. = 4^44 15* .o. R. A. Mean Sun at G. M. N. = 17"
46 40 S .38.
* Time-balls are now in use in the principal ports on the Atlantic, Pacific,
and Gulf coasts and on the Great Lakes.
138 PRACTICAL ASTRONOMY
6. Time by equal altitudes, Oct. 13, 1906.
R. A. Decl. Watch
vOphiuchi (W) ij h 53 52*.!$ -9 45' 34".6 7* 13"* 49*
t Ceti (E) o 14 40 .99 9 20 25 .7 7 28 25
Lat. = 42 18'; long. = 4^ 45 o6*.8. R. A. of Mean Sun at G. M. N.=
13* 24 323.56.
7. Show by differentiating Equa.  that the most favorable position of the
sun for a time observation is on the prime vertical. The differential coefficients
-jT- and -r= should be a minimum to give the greatest accuracy. The expres-
sions obtained may be simplified by means of Equa. .
8. Compute the watch correction from the observation given on p. 124. The
R. A. of the mean sun on Jan. 9, 1907, was 19^ n m 29 8 .49,
OBSERVATIONS FOR LONGITUDE
85. Method of Measuring Longitude.
The measurement of the difference in longitude of two places
depends upon a comparison of the local times of the places at
the same absolute instant of time. One important method
is that in which the timepiece is carried from one station to
the other and its error on local time determined in each place.
The most precise method, however, and the one chiefly used
in geodetic work, is the telegraphic method, in which the local
times are compared by means of electric signals sent through a
telegraph line. Other methods, most of them of inferior accu-
racy, are those which depend upon a determination of the moon's
position (moon culminations, eclipses, occultations) and upon
eclipses of Jupiter's satellites, and those in which terrestrial
signals are employed.
86. Longitude by Transportation of Timepiece.
In this method the error of the watch or chronometer with
reference to the first meridian is found by observing the local
time at the first station. The rate of the timepiece should be
determined by making another observation at the same place
at a later date. The timepiece is then carried to the second
station and its error determined with reference to this meridian.
If the watch runs perfectly the two watch corrections will
differ by just the difference in longitude. Assume that the first
observation is made at the easterly station and the second at
the westerly station. To correct for rate, let r be the daily
rate in seconds, + when losing when gaining, c the watch
correction at the east station, c' the watch correction at the
west station, d the number of days between the observations,
and T the watch reading at the second observation. Then the
difference in the longitude is found as follows :
Local time at W. station = T + c'
Local time at E. station = T + c + dr
Diflf. in time = Diff. in Long. = c-\- dr c' . [i]
The same result will be obtained if the stations are occupied
in the reverse order.
If the error of a mean-time chronometer or watch is found
by star observations, it is necessary to know the longitudes
accurately enough to correct the sun's right ascension. If a
sidereal chronometer is used and its error found on L. S. T. this
correction is rendered unnecessary.
In order to obtain a check on the rate of the timepiece the
observer should, if possible, return to the first station and again
determine the local time. If the rate is uniform the error in
its determination will be eliminated by taking the mean of the
results. This method is not as accurate as the telegraphic
method, but if several chronometers are used and several round
trips between stations are made it will give good results. It is
useful at sea and in exploration surveys.
Observations for local mean time at meridian A indicate
that the watch is 15 40* slow. At a point B, west of A, the
watch is found to be 14 io s slow on local mean time. The
watch is known to be gaining 8 s per day. The second obser-
vation is made 48 hours after the first. The difference in longi-
tude is therefore
+ I5 m 40 8 - 2 X 8 s - i4 m io s = i* i 4 s .
The meridian B is therefore i m 14" or 18' 30" west of meridian A.
87. Longitude by the Electric Telegraph.
In the telegraphic method the local sidereal time is accurately determined by
star transits observed at each of the stations. The observations are made
with large portable transits and are recorded on chronographs which are connected
OBSERVATIONS FOR LONGITUDE 141
with break-circuit chronometers. The stars observed are chosen in such a manner
as to determine the errors of the instruments so that these may be eliminated
from the results as completely as possible. Some of the stars are slowly moving
(circumpolar) stars and others are more rapidly moving stars near the zenith;
a comparison of these two makes it possible to compute the azimuth of the line
of collimation. Half of the stars are observed with the instrument in one position,
half in the reversed position; this determines the error in the sight line. The
inclination error is measured with the striding level.
After the corrections to the two chronometers have been accurately determined
the two chronographs are switched into the main-line circuit and signals are sent
by breaking the circuit a number of times by pressing a telegraph key. These
signals are recorded on both chronographs. In order to eliminate the error due
to the time required in transmitting a signal,* these signals are sent first in one
direction (E-W) and afterward in the opposite direction (W-E). In this
manner the transmission time is eliminated, provided it is constant. The personal
errors of the observers are eliminated by the observers exchanging places in the
middle of the series; i.e., the above operation would be repeated for about five
nights with the observers in one position and then for five nights after the observers
have exchanged positions. After all of the observations have been corrected for
instrumental errors, and the error of the chronometer on local sidereal time is
known, each signal sent over the main line will be found to correspond to a certain
instant of sidereal time at the east station and a different instant of sidereal time
at the west station. This difference is the difference in longitude. The mean of
all these values is the final difference free from errors in transmission time and per-
sonal errors. By this method the difference in longitude may be determined with
an error of perhaps 10 to 20 feet on the earth's surface.
88. Longitude by Transit of the Moon.
A method which is easily used with the surveyor's transit and which, although
not precise, may be of use in exploration surveys, is that of determining the moon's
right ascension by observing its transit over the meridian. The right ascension
of the moon's centre is tabulated in the Nautical Almanac for every hour of
Greenwich Mean Time; hence, if the right ascension can be determined, the
Greenwich time can be computed. A comparison of this with local time gives
The observation consists in placing the instrument in the plane of the meridian
and noting the time of transit of the bright limb f of the moon and also of several
stars whose declinations are nearly the same as that of the moon. The observed
time interval between the moon's transit and that of a star (reduced to sidereal
time if necessary), added to or subtracted from the star's right ascension, gives
the right ascension of the moon's limb. A value of the right ascension is obtained
* In a test made in 1905 it was found that the time signal sent from Washington
reached Lick Observatory, Mt. Hamilton, Cal., in 0^.05.
t The table of moon culmination in the Ephemeris shows which limb (I or II)
may be observed. See also note, p. 143.
142 PRACTICAL ASTRONOMY
from each star and the mean value used. To obtain the right ascension of the
centre of the moon it is necessary to apply to the right ascension of the limb a
correction taken from the Ephemeris called " sidereal time of semidiameter passing
meridian." In computing this correction the increase in the right ascension
during this short interval has been allowed for; so the result is not the right ascen-
sion of the centre at the instant of the observation, but its right ascension at the
instant of the transit of the centre over the meridian. If the west limb was
observed this correction must be added; if the east limb was observed it must be
subtracted. The result is the right ascension of the centre at the instant of
transit, which is also the local sidereal time at that instant. Then the Greenwich
Mean Time corresponding to this instant is found by interpolating in the table
giving the moon's right ascension for every hour. To obtain the G. M. T. by
simple interpolation find the next less right ascension in the table and the "diff.
for i m " on the same line; subtract the tabular right ascension from the given
right ascension (found from the observation) and divide this difference by the
" diff. for I TO ." The result is the number of minutes and decimals of minutes
to be added to the hour of G. M. T. opposite the tabular right ascension used.
If the " diff. for i m " is varying rapidly it will be more accurate to interpolate as
follows. Interpolate between the two values of the " diff. for i m " and obtain a
" diff. for I TO " which corresponds to the middle of the interval over which the inter-
polation is carried. In observations made with the surveyor's transit this more
accurate interpolation is seldom necessary.
In order to compare the Greenwich time with the local time it is necessary to
convert the G. M. T. just obtained into the corresponding instant of Greenwich
Sidereal Time. The difference between this and the local sidereal time is the longi-
tude from Greenwich.
In preparing for observations of the moon's transit the Nautical Almanac
should be consulted (Table of Moon Culminations) to see whether an observation
can be made and to find the approximate time of transit. The civil date should
be converted into astronomical before entering the Almanac. The time of the
moon's transit may be taken from the column headed " Mean time of transit "
and corrected for longitude, or it may be computed from the approximate right
ascension. The altitude of the moon should be computed as for a star, and
in addition the parallax correction should be applied. The moon's parallax
is so large that the moon probably would not be in the field of the telescope
at all if this correction were neglected. The horizontal parallax multiplied
by the cosine of the altitude is the correction to be applied; the moon will
appear lower than it would if seen at the centre of the earth, so the correction is
Since the moon increases its right ascension about 2 s in every i m of time it is
evident that any error in determining the right ascension will produce an error
about thirty times as great in the longitude, so that this method cannot be made to
give very precise results.
Following is an example of an observation for longitude by the method of
moon culminations made with an engineer's transit.
OBSERVATIONS FOR LONGITUDE
Observed transit of Moon on Jan. 9, 1900, for longitude. Moon's west b'mb
passed cross hair at 6 A 59"* 378.7; 5 Ceti passed at 7^ O2 m 578.0; and y Ceti passed
at 7* 06"* 428.0.
R. A. 5 Ceti
R. A. M.'s Limb
-jh 02 m 578.0 y Ceti
6 59 37-7 Moon's Limb
Time of s. d. passing merid. =
R. A. M.'s Centre
7* o6 m 42* . o
6 59 37 -7
= 03 19 . 85 Sid. int. = 07"* 05*. 46
= 2 34 23 . 02 R. A. 7 Ceti =2 38 08 . 77
31 038.17 R.A.M.'s Limb = 2 h 31"* 03*. 31
31 03 .31
03 s . 24
$2 12*. 10 = L. S. T.
G. M. T.
From the Nautical Almanac.
R. A. Moon
2 h 29 m 55*. 77
2 32 12 .32
Diff. for i"
32"* 12 s . 10
2 9 55 -77
2 m 168.33 = 1368.33 log 2. 13459 G. M. T. = n h 59" 528.86
Interpolated Diff . i m = 2.2767 logo. 35730 R.A.M.S. = 19 14 15 .92
i 52 .86
M. T. Interval = 59 TO .88i log 1. 77729
G. S. T. = 7 h i6 m 018.64
G. M. T. = i i h 59 m 528.86 L. S. T. =2 32 12.10
Long. W. = 4 h 43 49 s -54
NOTE. It has already been stated that the moon moves eastward on the celes-
tial sphere at the rate of about 13 per day; as a result of this motion the time of
meridian passage occurs about 51"* later (on the average) each day. On account
of the eccentricity of its orbit, however, the actual retardation may vary consid-
erably from the mean. The moon's orbit is inclined at an angle of about 5 08'
to the plane of the earth's orbit. The line of intersection of these two planes ro-
tates in a similar manner to that described under the precession of the equinoxes,
except that its period is only 19 years. The moon's maximum declination, there-
fore, varies from 23 27'+ 5 08' to 23 27' - 5 08', that is, from 28 35' to 18 19',
according to the relative position of the plane of the moon's orbit and the plane of
the equator. The rapid changes in the relative position of the sun, moon, and earth,
and the consequent changes in the amount of the moon's surface that is visible
from the earth, cause the moon to present the different aspects known as the
moon's phases. Fig. 58 shows the relative positions of the three bodies at several
FIG. 58. THE MOON'S PHASES
different times in the month. The appearance of the moon as seen from the earth
is shown by the figures around the outside of the diagram.
It may easily be seen from the diagram that at the time of first quarter the
moon will cross the meridian at about 6 P.M. ; at full moon it will transit at mid-
night; and at last quarter it will transit at about 6 A.M. Although the part of the
illuminated hemisphere which can be seen from the earth is continually changing,
the part of the moon's surface that is turned towa-d the earth is always the same,
because the moon makes but one rotation on its axis in one lunar month. Nearly
half of the moon's surface is never seen from the earth.
OBSERVATIONS FOR LONGITUDE 145
Questions and Problems.
1. Compute the longitude from the following observed transits: 6 Aquarii,
$h i6 TO O4 S ; TT Aquarii, 5^ 24 40"; moon's W. limb, $ h 32 27*5 X Aquarii,
5 h 5i TO 47 s - R- A. Aquarii = 22* n m 2j s .6; R. A. ir Aquarii, = 22^ 2o m 04*. 6;
R. A. \ Aquarii = 22^47 i8 s .3; sidereal time of semidiameter passing meri-
dian = 6o*.3; at G. M. T. 10*, moon's R. A. = 22^ 27 53 s .s; diff. for i w =
i s .g8oo; R. A. mean sum at G. M. N. = i6 h 38 28*.o.
2. Can the longitude be computed by comparing G. M. T. and L. M. T.?
3. Which limb can be observed in a P. M. observation of a moon culmination ?
4. At about what time (mean local) will the moon transit at first quarter ?
OBSERVATIONS FOR AZIMUTH
89. Determination of Azimuth.
The determination of the azimuth of a line is of frequent
occurrence in the practice of the surveyor, and is the most
important to him of all the astronomical problems. On account
of the high altitudes of the objects observed, as compared with
those observed in surveying, the adjustments of the instru-
ment and the elimination of errors are of unusual importance
in these observations. All of the precautions mentioned in
Chapter VIII in regard to stability of the instrument, etc.,
should be carefully observed : the instrument should be allowed
to stand for some time before observations are begun; temper-
ature changes from any source, such as heat from the lamp or
from the hand, are to be avoided; the clamps and tangent screws
should be used with the same care as in triangulation work if
the greatest accuracy is desired in the results.
90. Azimuth Mark.
When the observation is made at night it is frequently incon-
venient to sight directly at the object whose azimuth is to be
determined ; it is necessary in such cases to determine the azimuth
of a special mark called the azimuth mark, which can be seen
both at night and in daylight, and then to measure the angle
between this mark and the first object during the day. The
azimuth mark usually consists of a lamp set inside of a box
having a small hole cut in the side, through which the light
may shine. The size of the opening should be determined by
the distance of the mark; for accurate work it should subtend
an angle not greater than about o".5 to i".o. If possible,
the mark should be a mile or more distant, so that the focus
of the telescope will not have to be altered when changing from
OBSERVATIONS FOR AZIMUTH 147
the star to the mark. It is frequently necessary, however,
to set the mark nearer on account of the topographic and other
91. Azimuth of Polaris at Elongation.
The simplest method of determining the direction of the
meridian with accuracy is by means of an observation of the
polestar, or any other close circumpolar, when it is at its greatest
elongation. (See Art. 19, p. 31.) The appearance of the
constellations at the time of this observation on Polaris may
be seen by referring to Fig. 49. When the polestar is west of
the pole the Great Dipper is on the right and Cassiopeia on the
left. The exact time of elongation may be found by comput-
ing the sidereal time when the star is at elongation, and convert-
ing this into mean solar time (local or standard) by the methods
of Arts. 34 and 35. To find the sidereal time of elongation first
compute the hour angle P by Equa.  and then convert it
into time. If western elongation is desired, then P is the hour
angle; if eastern elongation is desired, then 24^ P is the true
hour angle. The sidereal time is then found by Equa. .
An average value of P for Polaris for latitudes between 30
50 is about 5^55 m ; this is sufficiently accurate for computing
the time of elongation for many purposes. Approximate values
of the times of elongation of Polaris may be taken from Table V.
Find the Eastern time of Elongation of Polaris on April 6,
1904, in lat. 42 21'; long. 4 A 44 m i8 s W. The right ascension,
is i h 23 48 S .3; the declination is + 88 47' 43".6; the sun's
right ascension is o h 57 m 22^44.
log tan I = 9. 95977 P = 5 h 55 36". S S = ? h ig m 24". 8
log tan D = i. 67723 R = i 23 48.3 R s = o 58 09.1,
log cos P =8.28254 S = 7 h 19 24*. 8 6 h 2i m 15 s . 7
P = 88 54' 07" + C' = - i 02 . 5
= $h ([JOT 36 s . 5
M. L. T. = 6 h 20 i3 s . 2
E. S.T. = 6> 04"* 3i.a
148 PRACTICAL ASTRONOMY
The transit should be set in position half an hour or so
before elongation. The star is bisected by the vertical cross
hair, and as it moves out toward its greatest elongation its
motion is followed by means of the tangent screw of the upper
or the lower plate. Near the time of elongation the star will
appear to move almost vertically, so that no motion in azimuth
can be detected for five minutes or so before or after elonga-
tion. About 5 m before elongation, centre the plate levels, set
the cross hair carefully on the star, lower the telescope with-
out disturbing its azimuth, and set a stake or a mark carefully
in line at a distance of several hundred feet north of the transit.
Reverse the telescope, recentre the levels if necessary, bisect
the star again, and set another point beside the first one. If
there are errors of adjustment the two points will not coincide;
the mean of the two is the true point. The angle between the
meridian and the line to the stake (the star's azimuth) is found
by the equation
sin Z = sin p sec L 
where Z is the azimuth from the north; p, the polar distance of
the star; and L, the latitude of the place. L does not have to
be known with great precision; an error of i' in L produces
only about i" error in the azimuth of Polaris for latitudes within
the United States. The above method may be applied to any
close circumpolar star. For Polaris, whose polar distance is
about i 10', it is usually accurate enough to use the formula
Z" = p" sec L, 
in which Z" and p" are expressed in seconds of arc. This
computed angle may be laid off in the proper direction with a
transit (by daylight), using the method of repetitions, or with
a tape, by means of a perpendicular offset calculated from the
measured distance to the stake and the calculated azimuth
angle. (Fig. 59.) The result is the true north and south line.
It is often desirable to measure the horizontal angle between
OBSERVATIONS FOR AZIMUTH
the star at elongation and some fixed point instead of marking
the meridian itself. On account of the slow change in azimuth
there is ample time to measure several repetitions before the
error in azimuth amounts to more than i" or
2".* The errors of adjustment of the transit
will be eliminated if half of the angles are
taken with the telescope erect and half in-
verted. The plate levels should be recentred
for each position of the instrument before the
measurements are begun and while the telescope
is pointing toward the star.
Compute the azimuth of Polaris at greatest
elongation on April 6, 1904, in latitude 42
21' N. The declination of the star for the given
date is + 88 47' 43 // .6.
log sin p = 8. 32267
log sec L = o. 13133
log sin Z = 8. 45400
Z =1 37' 47". 9
By using the angles in place of the sines, neglecting fractions
of a second, the following result is obtained :
P = 4336"
log p = 3. 6371
log sec L = o. 1313
Z" = 5867
= i 37' 47"
92. Observations Near Elongation.
If the observation is made on Polaris at any time within half
an hour of elongation, the azimuth of the star at each pointing
* In latitude 40 the azimuth changes about i' in half an hour before or after
elongation; the change in azimuth varies approximately as the square of the time
of the telescope may be reduced to its value at elongation,
provided the time is known. The formula for this reduction is
C = 112.5 X 3600 X sin i" X tanZ e X (T - T e ) 2 * 
in which Z e is the azimuth at elongation; T, the observed time;
T e , the time of elongation ; and C, the correction in seconds of
* For the rigorous demonstration of this for-
mula, which is rather complex, see Doolittle's.
Practical Astronomy. The following proof,
although inexact, gives substantially the same
result. In Fig. 60 S is the position of Polaris
and E its position when at greatest elongation,
the angle SPE, or i, being not greater than
about 8. In the triangle SPM ,
tan MP = tan PS cos SPM
Since the arcs are small, we may put
Replacing cos i by the series i [->
MP = SP cos 5PM,
MP = p cos i.
EM = EP - PM
P ~~ P cos *
EM = p
In the triangle ZME, Z.E = 90, and ZM = ZP (nearly), whence
sin M ZE =
sin p i i* , . -.
= T Xp (nearly),
COS L 2
= sin Z e Xp->
in which Z e is the azimuth at elongation. Replacing sin MZE by its arc in seconds
(C") and reducing i to seconds of time,
C" = - X sin i" X (6o) 2 X (is) 2 X sinZ e .
Replacing sin Z e by tan Z e produces an error of only about o". 02 for Polaris in
latitude 40 and reduces  to .
OBSERVATIONS FOR AZIMUTH 151
arc. T T e must be in minutes of (sidereal) time. The
factor 112.5 x 36 x sin I/f x tanZ e may be computed, and
then all observations made at the same place at about the same
date may be reduced by multiplying the square of the time in-
tervals in minutes by the factor computed. Table VII gives
values of the factor for values of Z e ranging from i to 2.
These corrections will also be found in Table Via at the end
of the Nautical Almanac.
Three repetitions of the angle between Polaris at western elongation and a
mark supposed to be on the meridian, April 6, 1904. Lat. 42 21'; long. 71 O4'.5 W.
The observed times are 6* 28 m 30*, 6^ 31 2o s and 6^ 34 2o s . First reading of
vernier = o oo'; last reading of vernier = 4 51' oo". The R. A. of Polaris =
i h 2$ 488.3; its declination = + 88 47' 43".6. R. A. Mean Sun at G. M. N. =
o fc jj7 TO 22 S .44.
From this data the Eastern time of elongation is found to be 6 h 04 3i s .2.
The intervals (T - T e ) are 23 58^.8, 26 48 S .8 and 29 48*. 8. The azimuth
of the star at elongation is i 37' 48". From Table VII the factor is found to
be .0559. The resulting corrections are 32", 40" and 50". Adding these to the
third reading, the sum is 4 53' 02". One third of this is i 37' 41", the measured
angle between the mark and the star at elongation. The meridian mark is there-
fore 7" west of north, according to this observation.
93. Azimuth by an Altitude of the Sun.
In order to determine the azimuth of a line by means of an
observation on the sun the instrument should be set up over
one of the points marking the line and carefully levelled. The
plate vernier is first set at o and the vertical cross hair sighted
on the other point marking the line. The colored shade glass
is then screwed on to the eyepiece, the upper clamp loosened,
and the telescope turned toward the sun. The sun's disc should
be sharply focussed before beginning the observations. In
making the pointings on the sun great care should be taken
not to mistake one of the stadia hairs for the middle hair. If
the observation is to be made, say, in the forenoon (in the
northern hemisphere), first set the cross hairs so that the ver-
tical hair is tangent to the right edge of the sun and the hori-
zontal hair cuts off a small segment at the lower edge of the
disc. (Fig. 6 1.)* The arrow in the figure shows the direction of
the sun's apparent motion. Since the sun is now rising it will
in a few seconds be tangent to the horizontal hair. It is only
necessary to follow the right edge by means of the upper plate
tangent screw until both cross hairs are
tangent. At this instant, stop following
the sun's motion and note the time. If
it is desired to determine the time accu-
rately, so that the watch correction may be
found from this same observation, it can
be read more closely by a second observer.
FIG. 61. POSITION OF Both the horizontal and the vertical
SUN'S Disc A FEW SECONDS circles are read, and both angles and the
BEFORE OBSERVATION t j me are rec orded. The same observa-
(A. M. Observation in Northern . . . , .
Hemisphere.) tion may be repeated three or four
times to increase the accuracy. The instrument should then
be reversed and the set of observations repeated, except that
the horizontal cross hair is set tangent to
the upper edge of the sun and the ver-
tical cross hair cuts a segment from the
left edge (Fig. 62). The same number
of pointings should be taken in each
position of the instrument. After the
pointings on the sun are completed the
telescope should be turned to the mark FlG 62 p osmc >N OF
again and the vernier reading checked. SUN'S Disc A FEW SECONDS
If the transit has a vertical arc only, the BEFORE OBSERVATION
telescope cannot be used in the reversed (A- M ' Hembphere a ) No
position and the index correction must therefore be determined.
If the observation is to be made in the afternoon the positions
will be those indicated in Fig. 63.!
* In the diagram only a portion of the sun's disc is visible; in a telescope of low
power the entire disc can be seen.
t It should be kept in mind that if the instrument has an inverting eyepiece
the direction of the sun's apparent motion is reversed. If a prism is attached to
the eyepiece, the upper and lower limbs of the sun are apparently interchanged,
but the right and left limbs are not.
OBSERVATIONS FOR AZIMUTH
In computing the azimuth it is customary to neglect the cur-
vature of the sun's path during the short interval between the
first and last pointings, unless the series extends over a longer
period than is usually required to make such observations.
If the observation is taken near noon the curvature is greater
than when it is taken near the prime vertical. The mean of
the altitudes and the mean of the horizontal angles are assumed
to correspond to the position of the sun's centre at the instant
shown by the mean watch reading. The mean altitude read-
ing corrected for refraction and parallax is the true altitude of
FIG. 63. POSITIONS OF SUN'S Disc A FEW SECONDS BEFORE OBSERVATION
(P. M. Observation in Northern Hemisphere.)
the sun's centre. The azimuth is then computed by any one
of the formulae on page 34. The resulting azimuth combined
with the mean horizontal circle reading gives the azimuth of
the mark. Five-place logarithmic tables will give the azimuth
within 5" to 10", which is as precise as the azimuth can be
determined by this method.
If for any reason only one limb of the sun has been observed,
the azimuth observed may be reduced to the centre of the sun
by applying the correction 5 sec h, where S is the semidiameter
and h is the altitude of the centre.
If one has at hand a set of tables containing log versed sines (such as are in-
cluded in railroad engineering tables) the following formulae will sometimes be
cos (L + h) + sin D 
vers Z s =
cos L cos h
cos (L h) sin D
versZ n = - =-^ -.
cos L cos h
The sum or difference in the numerator must be computed by natural functions
and the remainder of the work performed by means of logarithms.
Observation on Sun for Azimuth.
Lat. 42 21' N. Long. 4* 44"* 18* W. Date, Nov. 28, 1905.
Hor. Circle. Vert. Circle. Watch.
Ver. A. B. A. M.
R&Llimbs 311 48 48.5 14"
R & L limbs 312 20 20 15
L & U limbs 312 27
L & U limbs 312 52
Mark 238 14
8 42 19
Mean reading on
Mean reading on
Mark N. of sun
L = 42 21'. o
h = 15 22 .7
p = in 15.7
238 14'. o
74 07'. 7
15 22'. 7
Mean = 8 h 43 47*
G. M. T =
Sun's Decl. at G. M. N. =- 21
- 2 6".8i X i m .73
14' 54". 4
- 46 . 4
2 5 = 168 59'. 4
s = 84 29'. 7
s L = 42 08'. 7
5 h = 69 07 . o
s p = 26 46 . o
s = 84 29.7
Declination = 21 15' 40". 8
N. Polar distance = m 15' 40". 8
log sin 9. 82673
log sin 9.97049
log sec o. 04922
log sec 1.01804
log tan \Z n = o. 43 22 4
\ Z n = 69 42'. 9
Zn = 139 25'. 8
Mark N. of sun = 74 07 . 7
Bearing of Mark= N 65 18'. i E
By differentiating Equa.  it may be shown that when the latitude is greater
than the sun's declination the greatest accuracy in the azimuth, so far as errors
in altitude are concerned, is secured when the sun is somewhere between the prime
vertical and the six-hour circle; the exact position for maximum accuracy depends
upon the latitude and upon the parallactic angle. If an observer were on the
equator and the sun's declination zero, the motion would be vertical and the
change in azimuth would be zero. In the preceding example the azimuth increases
about i' 50" for an increase of i' in the altitude. Errors in the azimuth due to
errors in the assumed value of the latitude are a minimum when the sun is on the
six-hour circle. Observations very near the horizon, however, are subject to errors
OBSERVATIONS FOR AZIMUTH 155
in the refraction, since the tabular values of the mean refraction may be largely
in error for very low altitudes under the temperature and pressure conditions
existing at the time of the observation. The general rule is therefore to avoid
observations near the meridian and also those within 10 or less of the horizon.
If it is desired to compute the hour angle of the sun from the same observations
used in determining the azimuth, it may be found by formula , in which case
no new logarithms have to be taken from the tables; or it may be found by the
sin P = sinZ cos h sec D. 
The value of P and the error of the watch obtained by the use of this formula
are given below.*
log sinZ = 9. 81317
log cos h = 9. 98416
log sec D = o. 03061
log sin P = 9. 82794
P = 42 17' 26"
= 2^ 49 m 09* . 7
L. A. T. =9 10 50 .3
Eq. T. = 12 02 .8
L. M. T. = S h 58 47*. 5
E. S.T. = 8* 43 TO 5 s -5
Watch =8 43 47 . 2
Watch fast = 4i s - 7
94. Azimuth by an Altitude of a Star.
The method described in the preceding article applies equally
well to an observation on a star, except that the star's image is
bisected with both cross hairs and the parallax and semidiameter
corrections become zero. The declination of the star changes so
little during one day that it may be regarded as constant, and
consequently the time of the observation is not required. Errors
in the altitude and the latitude may be partially eliminated by
combining two observations, one on a star about due east and
the other on one about due west.
95. Azimuth Observation on a Circumpolar Star at any Hour Angle.
The most precise determination of azimuth may be made by measuring the
horizontal angle between a circumpolar star and an azimuth mark, the hour angle
* See also Art. 102, p. 166, and Art. no, p. 175.
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.  and the azimuth of the star at the instant
may be computed. Since the close circumpolar stars move very slowly and
errors in the observed times will have a small effect upon the computed azimuth,
it is evident that only such stars should be used if precise results are sought. The
advantage of observing the star at any hour angle, rather than at elongation, is
that the number of observations may be increased indefinitely and greater accuracy
The angles may be measured either with a repeating instrument (like the
engineer's transit) or with a direction instrument in which the circles are read with
great precision by means of micrometer microscopes. For refined work the instru-
ment should be provided with a sensitive striding level. If there is no striding
level provided with the instrument* the plate level which is parallel to the hori-
zontal axis should be a sensitive one and should be kept well adjusted. At all
places in the United States the celestial pole is at such high altitudes that errors
in the adjustment of the horizontal axis and of the sight line have a compara-
tively large effect upon the results.
The star chosen for this observation should be one of .the close circumpolar stars
given in the special list in the Nautical Almanac. (See Fig. 64.) Polaris is the only
bright star in this group and should be used in preference to the others when it is
* The error due to inclination of the axis may be eliminated by taking half
of the observations direct and half on the image of the star reflected in a basin
OBSERVATIONS FOR AZIMUTH 157
practicable to do so. If the time is uncertain and Polaris is near the meridian,
in which case the computed azimuth would be uncertain, it is better to use 51
Cephei,* because this star would then be near its elongation and comparatively
large errors in the time would have but little effect upon the computed azimuth.
If a repeating theodolite or an ordinary transit is used the observations consist
in repeating the angle between the star and the mark a certain number of times
and then reversing the instrument and making another set containing the same
number of repetitions. Since the star is continually changing its azimuth it
is necessary to read and record the time of each pointing on the star with the
vertical cross hair. The altitude of the star should be measured just before and
again just after each half-set so that its altitude for any desired instant may be
obtained by simple interpolation. If the instrument has no striding level the
cross-level on the plate should be recentred before the second half-set is begun.
If a striding level is used the inclination of the axis may be measured, while the
telescope is pointing toward the star, by reading both ends of the bubble, with the
level first in the direct position and then in the reversed position.
In computing the results the azimuth of the star might be computed for each
of the observed times and the mean of these azimuths combined with the mean
of the measured horizontal angles. The labor involved in this process is so great,
however, that the practice is first to compute the azimuth corresponding to the
mean of the observed times, and then to correct this result for the effect of the
curvature of the star's path, i.e., by the difference between the mean azimuth
and the azimuth at the mean of the times. The formula for the azimuth is
z _ sinP
cos L tan D sin L cos P
The formula given below, although not exact, is sufficiently accurate for all work
except refined geodetic observations.
Z" = p" sin P sec h, 
in which Z" and p" are in seconds of arc. In this formula the arcs have been
substituted for their sines. The precision of the computed azimuth depends
chiefly upon the precision with which h can be determined. If the vertical arc
cannot be relied upon, and the latitude is known accurately, the first formula
may be preferred. If desired, the altitude of Polaris may be computed by formula
 and its value substituted in .
* 51 Cephei may be found by first pointing on Polaris and then changing the
altitude and the azimuth by an amount which will bring 51 Cephei into the field.
The difference in altitude and in azimuth may be obtained with sufficient accuracy
by holding Fig. 64 so that Polaris is in its true position with reference to the
meridian (as indicated by the position of S Cassiopeia) and then estimating the
difference in altitude and the difference in azimuth. It should be remembered
that the distance of 51 Cephei east or west of Polaris has nearly the same ratio
to the difference in azimuth that the polar distance of Polaris has to its azimuth
at elongation, i.e., i to sec L.
158 PRACTICAL ASTRONOMY
96. The Curvature Correction.
If we let Ti, Tz, T 3> etc. = the observed times, T = the mean of these times,
Zi, Zz, Z 3 , etc, = corresponding azimuths, and Z the azimuth at the instant T ,
Zi + Z 2 + Z n =Zo _ ^^ [Q j I 2 (T _ n)2 * [iQ7]
The quantity in brackets is the logarithm of a constant; 2 (T T ) 2 is the sum of
the squares of the time-intervals (in minutes and decimals) reduced to sidereal
intervals. The azimuth is therefore computed by first finding Z by Equa. 
and then correcting it by means of the last term of Equa. .
If it is desired to express ( T T ) in seconds of time the constant log becomes
[6.73672]. When the star is near culmination the curvature correction is very
small; near elongation it is a maximum.
97. The Level Correction.
The inclination i of the axis as determined by the striding level is given in
seconds of arc by
i = [(w + a/) -._(.+ e')] - , 
where w and e are the readings of the west and east ends of the bubble for the
direct position, and w' and e' are the same for the reversed positions, and d is the
angular value of one division of the level scale. The correction to the measured
horizontal angle is
C = i tan h. 
If the west end of the axis is too high (i positive) the telescope has to be turned
too far west in pointing at the star; the correction must therefore be added to the
measured angle if the mark is west of the star, subtracted if east. If the instru-
ment has no striding level the error must be eliminated as completely as possible
by relevelling between the half-sets.
98. Diurnal Aberration.
Strictly speaking, the computed azimuth of the star should be corrected for
diurnal aberration, the effect of which is to make the star appear farther east
than it actually is, because the observer is being carried due east by the diurnal
motion of the earth. The correction is
cos L cos Z
" 3I9 X cos h [IIO]
For all but the most precise observations it may be taken as o".32, since the factor
cos L cos Z . , ,
= is never far from unity.
*For the derivation of the formula see Doolittle's Practical Astronomy and
Hayford's Geodetic Astronomy.
OBSERVATIONS FOR AZIMUTH
RECORD OF AZIMUTH OBSERVATIONS
Instrument (B. & B. No. 3441) at South Meridian Mark. Boston, May 16, 1910.
(One division of level = i5".o.)
II A 2 4 "35.o
o oo' oo"
Corr. = 1 2". 5
Alt. Polaris at
jjA ^m 2O s -5 =
41 20' 30"
Alt. Polaris at
n& 51"* 04 s . o =
Mark. . .
*39 33' 3o"
41 i 8' 40"
66 35' 35"- o
II 42 45.5
39 33' 30"
44 09 . o
Corr. = 1 6". 5
Mark. . .
* 7 8 27' 30"
66 28' 59". 2
Alt. Polaris at
i2 h og m 31*. 5 =
41 15' 4o"
* Passed 360.
l6o PRACTICAL ASTRONOMY
RECORD OF TIME OBSERVATIONS
Polaris: Chronometer, 12*09"* 31*. 5; alt., 41 15' 40"
e Corvi: Chronometer, 12 13 37 .5; alt., 25 34 oo
Polaris: R. A. = i h 25 si s .i; decl. = +88 49' 24". 8
/Corvi: R. A. = i2 h 5"* 30*. 5; decl. = 22 07' 21". o
Chronometer R. A. Decl.
a Serpentis (E) i2 h 24 15*. 7 15*39"* Si 8 - 6 +6 42' 20". 7
f Hydra: (W) 12 18 32 .o 8 42 oo . 5 + 6 44 58 . 9
(Lat. = 42 21' oo" N.; Long. = 4* 44 i8. o W.)
From these observations the chronometer is found to be io w 22*. i fast.
COMPUTATION OF AZIMUTH
Mean of Observed times = n* 37"* 25*. 6
Chronometer correction = 10 22.1
Sidereal time = n 27 03 . 5
R. A. of Polaris = i 25 51 . i
Hour Angle of Polaris = 10 01 12 .4
P = 150 18' 06"
log cos L = 9. 868670
log tan D = i. 687490
log cos L tan D = 1.556160
cos L tan D = 35. 9882
log sin L = 9. 82844
log cos P = 9. 93884
log sin L cos P = 9. 76728
sin L cos P = . 5852
denominator = 36. 5734
log sin P = 9. 694985
log denom. = 1.563165
log tan Z = 8. 131820
Z = o 4 6' 34". 2
Curvature correction = 2. i
Azimuth of star = o 46 32.1
Measured angle, first half = 66 35' 35". o
Level correction = 12.5
Corrected angle = 66 35 22.5
Measured angle, second half = 66 28 59 . 2
Level correction = +16 .5
Corrected angle = 66 29 15 . 7
Mark east of star = 66 32 19 . i
Mark east of North = 65 45' 47". o
OBSERVATIONS FOR AZIMUTH 161
Observed altitudes of Regidus (east), Feb. n, 1908, in lat. 42 21'.
17 05' j h i2 m 16*
17 3i J 4 3i
17 49 l6 7
18 02 17 20
The right ascension of Regidus is 10* 03"* 29*. i; the declination is + 12 24' 57".
From these data the sidereal time corresponding to the mean watch reading
( 7 & I5 m 038.5) is found to be 4 h 53 42^.7.
Observed horizontal angles from azimuth mark to Polaris.
(Mark east of north.)
Telescope Direct Time of pointing on Polaris
Mark, o oo' 7 A 20 38*
Third repetition 201 48' 23 56
Mean= 67 16'. o -j h 22 318.3
Mark= o oo' 7 27 09
Third Repetition 201 54' 29 21
Mean = 67 18'. o 7 h 28 15 s . 7
Altitude of Polaris at f h 2o m 38* = 43 03'
Altitude of Polaris at 7 29 21 = 43 01
Mean watch reading for Polaris = j h 25 23*. 5
Corresponding sidereal time =5 04 04 . 4
Right Ascension of Polaris = i 25 32 .3
Hour-angle of Polaris =3 38 32 . i
P = 54 38'
P = 4251
log p = 3. 62849
log sin P = 9. 91141
log sec h = o. 13611
log azimuth = 3. 67601
azimuth = 4743"
= i 19'. o
Mean angle = 67 17 .o
Mark East of North = 65 58'. o
99. Meridian by Polaris at Culmination.
The following method is given in Lalande/s Astronomy and
was practiced by Andrew Ellicott, in 1785, on the Ohio and
Pennsylvania boundary survey. The direction of the meridian
is determined by noting the instant when Polaris and some
162 PRACTICAL ASTRONOMY
other star are in the same vertical plane, and then waiting a
certain interval of time, depending upon the date and the star
observed, when Polaris will be in the meridian. At this instant
Polaris is sighted and its direction then marked on the ground
by means of stakes. The stars selected for this observation
should be near the hour circle through the polestar; that is,
their right ascensions should be nearly equal to that
of the polestar, or else nearly i2 h greater. The stars
best adapted for this purpose at the present time are
d Cassiopeia and f Ursa Majoris.
The interval of time between the instant when
the star is vertically above or beneath Polaris and
)P the instant when the latter is in the meridian is
computed as follows : In Fig. 65 P is the pole, P' is
Polaris, S is the other star (8 Cassiopeia) and Z is
the zenith. At the time when S is vertically under
P' ', ZP'S is a vertical circle. The angle desired is
ZPP', the hour angle of Polaris. PP'*and PS, the
polar distances of the stars, are known quantities;
P' PS is the difference in right ascension, and may
be obtained from the Ephemeris. The triangle P'PS
may therefore be solved for the angle at P'. Sub-
tracting this from 180 gives the angle ZP'P; PP f
is known, and PZ is the colatitude of the observer.
The triangle ZP'P may then be solved for ZPP', the desired
angle. Subtracting ZPP' from 180 or i2 h gives the sidereal
interval of time which must elapse between the two
observations. The angle SPP' and the side PP' are so
small that the usual formulae may be simplified, by replacing
sines by arcs, without appreciably diminishing the accuracy
of the result. A similar solution may be made for the upper
culmination of 5 Cassiopeia or for the two positions of the
star f Ursa Majoris, which is on the opposite side of the
pole from Polaris. The above solution, using the right ascen-
sions and declinations for the date, gives the exact interval
OBSERVATIONS FOR AZIMUTH 163
required; but for many purposes it is sufficient to use a time
interval calculated from the mean places of the star and for a
mean latitude of the United States. The interval for the star
8 Cassiopeia for the year 1901 is 3 m .o; for 1910 it is 6 m .i, the
annual increase being o m .35. For f Ursa Majoris the interval
for 1901 is 3 m .7; for 1910 it is 6 m -7, the annual increase being
o m .33. Beginning with the issue for 1910 the American
Ephemeris and Nautical Almanac contains values of these
intervals (Table VII) for different latitudes and for different
dates. Within the limits of the United States it will generally
be necessary to observe on d Cassiopeia when Polaris is at
lower culmination and on f Ursa Majoris when Polaris is at
The determination of the instant when the two stars are in
the same vertical plane is necessarily approximate, since there is
some delay in changing the telescope from one star to the other.
The motion of Polaris is so slow, however, that a very fair
degree of accuracy may be obtained by first sighting on Polaris,
then pointing the telescope to the altitude of the other star (say
d Cassiopeia} and waiting until it appears in the field; when
d Cassiopeia is seen, sight again at Polaris to allow for its
motion since the first pointing, turn the telescope again to
5 Cassiopeia and observe the instant when it crosses the verti-
cal hair. The motion of the polestar during this short interval
may safely be neglected. The tabular interval of time corrected
to date must be added to the watch reading. When this com-
puted time arrives, the cross hair is to be set accurately on
Polaris and then the telescope lowered in this vertical plane and
a mark set in line with the cross hairs. The change in the
azimuth of Polaris in i m of time is not far from half a minute
of angle, so that an error of a few seconds in the time of sighting
at Polaris has but little effect upon the result. It is evident that
the actual error of the watch on local time has no effect what-
ever upon the result, because the only requirement is that the
interval should be correctly measured.
100. Azimuth by Equal Altitudes of a Star.
The meridian may be found in a very simple manner by means of two equal
altitudes of a star, one east of the meridian and one west. This method has the
advantage that the coordinates of the star are not required, so that the Almanac
or other table is not necessary. The method is inconvenient because it requires
two observations at night several hours apart. It is of special value to surveyors
in the southern hemisphere, where there is no bright star near the pole. The star
to be used should be approaching the meridian (in the evening) and about 3* or
4^ from it. The altitude should be a convenient one for measuring with the tran-
sit, and the star should be one that can be identified with certainty 6 h or S h later.
Care should be taken to use a star which will reach the same altitude on the oppo-
site side of the meridian before daylight interferes with the observation. In the
northern hemisphere one of the stars in Cassiopeia might be used. The position
at the first (evening) observation would then be at A in Fig. 66. The star should
be bisected with both cross hairs and the altitude read and recorded. A note or
a sketch should be made showing which star is used. The direction of the star
should be marked on the ground, or else the horizontal angle measured from some
reference mark to the position of the star at the time of the observation. When
the star is approaching the same altitude on the opposite side of the meridian
(at B) the telescope should be set at exactly the same altitude as was read at the
first observation. When the star comes into the field it is bisected with the ver-
tical cross hair and followed in azimuth until it reaches the horizontal hair. The
motion in azimuth should be stopped at this instant. Another point is then set
on the ground (at same distance from the transit as the first) or else another angle
OBSERVATIONS FOR AZIMUTH 165
is turned to the same reference mark. The bisector of the angle between the two
directions is the meridian line through the transit point. It is evident that the
index and refraction errors are eliminated, because they are alike for the two
observations. If one observation is made with the telescope direct and the other
with the telescope reversed, the other instrumental errors will be eliminated. Care
should be taken to level the instrument just before the observations. The accu-
racy of the final result may be increased by observing the star at several different
altitudes and using the mean value of the different results.
loz. Observation for Meridian by Equal Altitudes of the Sun in the Forenoon
and in the Afternoon.
This observation consists in measuring the horizontal angle between the mark
and the sun when it has a certain altitude in the forenoon and measuring the
angle again to the sun when it has an equal altitude in the afternoon. Since the
sun's declination will change during the interval, the mean of the two angles will
not be the true angle between the meridian and the mark, but will require a small
correction. The angle between the south point of the meridian and the point
midway between the two directions of the sun is given by the equation
Correction = =. -> [m]
cos L sin P
in which d is the hourly change in declination multiplied by the number of hours
elapsed between the two observations, L is the latitude, and P is the hour angle
of the sun, or approximately half the elapsed interval of time. The correction
depends upon the change in the declination, not upon its absolute value, so that
the hourly change may be taken with sufficient accuracy from the Almanac for
any year for the corresponding date.
In making the observation the instrument is set up at one end of the line whose
azimuth is to be determined, and the plate vernier set at o. The vertical cross
hair is set on the mark and the lower clamp tightened. The sun glass is then put
in position, the upper clamp loosened, and the telescope pointed at the sun.
It is not necessary to observe on both edges of the sun, but is sufficient to sight,
say, the lower limb at both observations, and to sight the vertical cross hair on
the opposite limb in the afternoon from that used in the forenoon. The hori-
zontal hair is therefore set on the lower limb and the vertical cross hair on the left
limb. When the instrument is in this position the time should be noted as accurately
as possible. The altitude and the horizontal angle are both read. In the after-
noon the instrument is set up at the same point, and the same observation is made,
except that the vertical hair is now sighted on the right limb; the horizontal hair
is set on the lower limb as before. A few minutes before the sun reaches an alti-
tude equal to that observed in the morning the vertical arc is set to read exactly
the same altitude as was read at the first observation. As the sun's altitude de-
creases the vertical hair is kept tangent to the right limb until the lower edge
of the sun is in contact with the horizontal hair. At this instant the time is again
noted accurately; the horizontal angle is then read. The mean of the two circle
readings, corrected for the effect of change in declination, is the angle from the
1 66 PRACTICAL ASTRONOMY
mark to the south point of the horizon. The algebraic sign of the correction is
determined from the fact that if the sun is going north the mean of the two ver-
nier readings lies to the west of the south point, and vice versa. The precision
of the result may be increased by taking several forenoon observations in suc-
cession and corresponding observations in the afternoon.
Lat. 42 18' N. Apr. 19, 1906.
A.M. Observations. P.M. Observations.
Reading on Mark, o oo' oo" Reading on Mark, o oo' oo"
( Alt., 24 58' ( Alt., 24 58'
U & L limbs < Hor. Angle, 357 14' 15" U & R limbs ] Hor. angle, 162 28'oo"
( Time, ; A 19"* 30* ( Time 4* i2 m 15*
5 elapsed time = 4^ 26 22 s
p = 66 35' 30" Incr. in decl. = + 52" X 4*. 44
log sin P = 9.96270 = + 230". 9
log cos L = 9.86902
9.83152 Mean Circle Reading =79 51' 08"
log 230". 9 = 2.36342 Correction = 5 40
2.53170 True Angle = S 79 45' 28" E.
Corr. = 340". 2 Azimuth = 280 14' 32"
1 02. Azimuth of Sun near Noon.
The azimuth of the sun near noon may be determined by means of Equa. ,
provided the local apparent time is known or can be computed. If the longitude
and the watch correction on Standard Time are known within one or two seconds
the local apparent time may be readily calculated. This method may be useful
when it is desired to obtain a meridian during the middle of the day, because the
other methods are not then applicable.
If, for example, an observation has been made in the forenoon from which a
reliable watch correction may be computed, then this correction may be used in
the azimuth computation for the observation near noon; or if the Standard Time
can be obtained accurately by a comparison at noon and the longitude can be
obtained from a map within about 1000 feet, the local apparent time may be
found with sufficient accuracy. This method is not usually convenient in mid-
summer, on account of the high altitude of the sun, but if the altitude is not
greater than about 50 the method may be used without difficulty. The obser-
vations are made exactly as in Art. 93, except that the time of each pointing is
determined more precisely; the accuracy of the result depends very largely upon
the accuracy with which the hour angle of the sun can be computed, and great
care must therefore be used in determining the time, The observed watch read-
ing is corrected for the known error of the watch, and is then converted into local
apparent time. The local apparent time converted into degrees is the angle at
the pole, P. The azimuth is then found by the formula
sin Z = sin P sec h cos D. 
Errors in the time and the longitude produce large errors in Z, so this method
should not be used unless both can be determined with certainty.
OBSERVATIONS FOR AZIMUTH
Observation on the sun for azimuth.
Lat. 42 21'. Long. 4* 44 i8 s W. Date, Feb. 5, 1910.
Hor. Circle. Vert. Circle. Watch.
Mark, o oo' (3O S fast)
app. L & L limbs, 29 01 31 49' ii' l 43 wl 22 s
app. U & R limbs, 28 39 31 16 n 44 20
Mean, 28 50' 31 32'. 5 n>>
Refr., i . 6 Watch corr. =
43 5i s
^ = 3 I ,3 / -9 E. S. T. =11*
D = - 1 6 06' 04". 5
43" 2i s
LA/T T TT^
S9 m 03 s
D = 16 02' 32". 2 Eq. t. =
Eq. t. =i4o8 s . 05 L. A. T. = n'
.217 X 4 h - 7 i .02 P =
1 44 53.
I ^m 6s
Eq. t. =
09 s . 07
log sin P = 8. 81847
log cos D = 9. 98275
log sec h = o. 06930
log sinZ = 8. 87052
Z = 4 15'. 4
Hor. Circle = 28 50
Azimuth = S 33 05'. 4 E
= 326 54'. 6
103. Combining Observations.
From the foregoing descriptions of field methods of observing, it will be seen
that but few of these methods are quite independent of the data obtained by other
observations, and in the practice of the engineer it often happens that no one of
the quantities which he desires can be completely determined until some or all
of the others are known approximately. The latitude may be determined directly
by means of a star at culmination, but it may be inconvenient or impracticable
in many cases to wait until either Polaris or a southern star comes to the meridian.
In all of the methods of determining time it is necessary to know either the latitude
or the direction of the meridian before the time can be directly computed. In all
of the methods of determining azimuth either the time, the latitude, or both must
be known. Where all of these quantities are entangled it is usually necessary to
obtain the true values by a series of approximations. In most cases, however,
very few approximations are necessary to give the greatest accuracy afforded by
If it is necessary to determine a precise azimuth, and nothing whatever is known
in regard to the latitude of the point or the local time, then all three may be accu-
rately determined by making observations of transits across the vertical circle
1 68 PRACTICAL ASTRONOMY
through Polaris, measuring the altitudes of all the stars, and then repeating the
horizontal angle between Polaris and an azimuth mark. The measured altitudes
of Polaris and the time-star make it possible to compute the sidereal time by two
approximations (Art. 78) without knowing the latitude. When the time is known
the latitude may be found by Arts. 68 and 69. If the instrument has only a
vertical arc (180), then the altitudes of the southern stars may be measured and
the first approximation to the latitude found from these observations. The alti-
tudes of Polaris may then be calculated closely enough for computing Equa. .
After the time and the latitude are known the azimuth is found directly. By
using the instrument in the two positions and increasing the number of obser-
vations the precision of all of the results may be increased.
The same results may be obtained by using the method of equal altitudes
(Arts. 80-82), combined with measured altitudes of the polestar and observations
for azimuth. By selecting a pair of stars having a large difference in right ascen-
sion or a small difference in declination, the time may be fairly well determined
by using an estimated latitude obtained by estimating a correction to the observed
altitude of Polaris. When the time is known approximately, a new value of the
latitude may be obtained, and with this new latitude the time may be recomputed.
The azimuth may then be found as before.
A very rapid but not very precise way of determining these three quantities
and also checking the azimuth is to sight on the mark, then to sight on the pole-
star, reading both the horizontal and vertical angles, and finally to sight on a prime
vertical star, reading both angles. Using an estimated latitude the PZS triangle
may be solved for P; with this value of P a close value of the latitude is found,
and the hour angle is then recomputed. If the latitude and the time are known
the azimuth may be determined from the polestar and checked by the azimuth
from the star near the prime vertical.
It is well when determining azimuths for surveying purposes to obtain checks
by methods which are independent of one another. For example, if the azimuth
is being found by angles measured to Polaris, a check may be obtained by turning
an angle from some star near the prime vertical (Art. 77) and measuring its alti-
tude simultaneously. Observations made on both east and west stars will increase
the accuracy. The azimuth thus computed is inferior in accuracy to that found
from Polaris, but the fact that it is independent makes it a valuable check against
mistakes or large errors in the Polaris observations. A sun observation made late
in the afternoon may be used in a similar way to check an evening observation
Questions and Problems
1. What error is caused by making the approximations in deriving formula
2. Derive formula .
3. Show that if the declination is less than the latitude the most favorable
conditions for determining azimuth by an altitude of the sun occur when the sun
OBSERVATIONS FOR AZIMUTH 169
is between the six-hour circle and the prime vertical. For greatest accuracy
j 7 /77
and TF should be a minimum. Differentiate Equa.  and simplify by means
of  and .
4. Show that the factor cos L cos Z sec h (Equa. [no]) is always nearly equal
5. Compute the approximate local mean time of eastern elongation of Polaris
on Sept. 10. R. A. of Polaris, i h 25. See Art. 63, p. 101, for an approximate
method of finding the R. A. of the mean sun. Use 5^ 55 for the hour angle of
Polaris at elongation (see Art. 91, p. 147).
6. Observation on sun May 15, 1906, for azimuth. Vernier A, on mark, read
o oo'. On the sun, right and lower limbs, vertical circle read 43 36'; vernier A
read 168 59' (right-handed); E. S. T., 2 h 52 45 s P.M. Upper and left limbs,
vernier A read 169 52'; vertical circle read 42 33'; E. S. T., 2 h 55^ 37* P.M. Dec-
lination at G. M. N. = + 18 42' 43". 6; diff. for i^ = + 3 5". 9 4. The latitude
of the place is 42 21' N; longitude 71 05' W. Compute the azimuth of the mark.
7. Compute the azimuth of Jupiter from the data given in Art. 77, p. 124.
8. Prove that the horizontal angle between the centre of the sun and the
right or left limb is 5 sec h where 5 is the apparent angular semidiameter and h is
the apparent altitude.
9. Prove that the level correction (Art. 97) is i tan h.
10. Why could not Equa.  be used in place of Equa.  in the method of
Art. 102 ?
n. If there is an error of 4 s in the assumed value of the watch correction and
an azimuth is determined by the method of Art. 102 (near noon), what would be
the relative effect of this error when the sun is on the equator and when it is 23
South? Assume the latitude to be 45 N. (See Table B.)
12. Make a set of azimuth observations by the method of Art. 93 (three point-
ings in each position of the instrument), and plot a curve using altitudes for ordinates
and horizontal angles for abscissae; also plot a curve using altitudes and times for
the two coordinates.
104. Observations at Sea.
The problems of determining a ship's position at sea and the
bearing of a celestial object at any time are based upon exactly
the same principles as the surveyor's problems of determining
his position on land and the azimuth of a line of a survey. The
method of making the observations, however, is different,
since the use of instruments requiring a stable support, such as
the transit and the artificial horizon, is not practicable at sea.
The sextant does not require a stable support and is well adapted
to making observations at sea. Since the sextant can be used
only to measure the angle between two visible points, it is
necessary to measure all altitudes from the sea-horizon and to
make the proper correction for dip.
Determination of Latitude at Sea
105. Latitude by Noon Altitude of Sun.
The determination of latitude by measuring the maximum
altitude of the sun's lower limb at noon is made in exactly the
same way as described in Art. 66. The observation should be
begun a little before local apparent noon and altitudes measured
in quick succession until the maximum is reached. In measur-
ing an altitude above the sea-horizon the observer should bring
the sun's image down 'until the lower limb appears to be in
contact with the horizon line. The sextant should then be
tipped by rotating right and left about the axis of the telescope
so as to make the sun's image describe an arc; if the lower limb
of the sun drops below the horizon at any point, the measured
altitude is too great, and the index arm should be moved until
the sun's image is just tangent to the horizon when at the lowest-
point of the arc. (Fig. 67.) This method is illustrated by the
Observed altitude of sun's lower limb 69 21' 30", bearing north. Index cor-
rection = i' 10"; height of eye = 18 feet; sun's declination at G. A. N. =
N 8 59' 32"; diff. i h = + 54"-43- Approx. lat. = 11 30' S; approx. long. =
i^ OO OT W.
Obs'd alt. = 69 21' 30'
Corr. = + 10 16
Alt. centre = 69 31' 46"
Declination = 9 oo 26
Colatitude = 78 32' 12"
Latitude = 11 27' 48" S
I. C. = - i' 10"
Dip = 4 12
r & p = o 20
S. D. = +i5 58
Corr. = + 10' 1 6"
8 59' 32"
9 oo' 26"
106. Latitude by Ex-Meridian Altitudes.
If for any reason the noon altitude has been lost, an altitude may be measured
near noon and this altitude corrected to the corresponding noon altitude by
Equa. . In order to make this " reduction to the meridian " it is necessary
to know the sun's hour angle. If the altitude is taken within a few minutes of
noon the reduction may be made by the more convenient formula, ; in practice
this is done by means of tables.
Observed altitude Jan. 20, 1910 = 20 05'; I. C. = o; G. M. T. = i* 35
28 s ; lat. by dead reckoning = 49 20' N; longitude by dead reckoning = i h 05
20*; height of eye = 16 feet; decl. at G. M. N. = 20 15' 02" S; diff. for i h
+ 32 ".o; S. D. = i6'i7".
G. M. T. =
Long. W =
35 m 2 gs
H. A. = cA 30*
= 7 32'
S. D. = + 16' 17'
I. C. = oo
Dip- = ~ 3 55
r & p 2 30
cos L = 9. 8140
cos D = 9. 9723
vers P = 7. 9361
Corr. = .0053
sin A = . 3461
Decl. G. M. N. =
Diff. for i h .i =
Decl. = 20 14' 27" S
Corr. = + 9' 52'
Obs. Alt. = 20 05
sin h m = . 3514
True Alt. = 20 14' 52" h m = 20 34' \
Decl. = 20 14 \
Colat. = 40 49'
Lat. = 49 n'
172 PRACTICAL ASTRONOMY
Determination of Longitude at Sea
107. By the Greenwich Time and the Sun's Altitude.
The usual method of finding the longitude at sea is to determine
the local mean time from an observed altitude of the sun (Art.
76) and to compare this with the Greenwich Mean Time as
shown by the chronometer. The error of the chronometer at
some previous date and its daily gain or loss are supposed to be
known. This is the same in principle as the method of Art. 86.
The value of the latitude used in solving the PZS triangle must
be that of the ship at the time the observation is made; this
latitude must be found by correcting the latitude by observation
at the previous noon for the run of the ship in the interval.
This is called the latitude by " dead reckoning." On account
of the large errors which may enter into this estimated latitude
it is important that the observation (" time-sight "). should be
made when the sun is near the prime vertical.
True alt. May 19, 1910 (P.M.) = 44 05'; G. M. T. = 6^ 55 io s . Lat. by
dead reckoning = 42 oo' N; decl. at G. M. N. = 19 38' 20" N; diff. i h =
-f- 32".?; equa. of time = 3 W 44*. i; deer, per i h = o s .i.
L = 42 oo' sec = .1289 Decl. G. M. N. 19 38' 20" N
D = 19 42 sec = . 0262 +32"_7 X 6^.9 = 3 46
cos = .9252 L D = 22 18 Cor'd. decl. = 19 42' 06" N
sin = . 6957 h - 44 05
diff. = . 2295 log = 9. 3608
log vers = 9.5159 Equa.G.M.N. = 3"* 44*. i
H. A. = 3^ n m 07 s o s .i X 6^.9 = . 7
Eq. t. = 3 43
Cor'd Eq. t. = 3 43*. 4
L. M. T. = 3 h 07 TO 24*
G. M. T. = 6 55 10
Long. W = 3^ 47 TO 46*
= 56 56' i W.
1 08. By a Lunar Distance.
The accuracy of the preceding method is wholly dependent
upon the accuracy of the chronometer giving the Greenwich
time. With steam vessels making short trips and carrying
NAUTICAL ASTRONOMY 173
several chronometers this method gives the longitude with
sufficient accuracy. In the days when commerce was carried
on chiefly by means of sailing vessels the voyages were of long
duration, and consequently the error of the chronometer could
be verified only at long intervals; furthermore, the chronom-
eters of that time were far less perfect than those of to-day, and
their rates were subject to greater irregularities. Under these
circumstances the method just described sometimes became
wholly unreliable; in such cases the
method of " lunar distance " was
used. Although this method is
necessarily of inferior accuracy it has
the advantage of being entirely inde-
pendent of the chronometer time. In
the Nautical Almanac previous to the
issue for 1912 there were given the
geocentric distances of the moon
from several bright stars, planets, and FlG 68
the sun, for every 3^ of Greenwich
Mean Time. If a lunar distance were measured at sea and this
distance reduced to the centre of the earth, the corresponding
instant of G. M. T. could be found by interpolation in these
The observation requires that the altitudes of the moon and
the sun or star should be measured simultaneously with the
distance, and that the chronometer should be read at the same
instant. In Fig. 68 let Z be the observer's zenith, M' the appar-
ent and M the true position of the moon, and S f and S the appar-
ent and true positions of the sun. The sun's apparent position
is higher than its true position because its refraction is greater
than its parallax. The moon's true position is higher than its
apparent position because the parallax correction is the greater.
The measured distance S'M' is to be reduced to the true dis-
tance SM. In the triangle ZS'M' the three sides have been
measured and the angle Z may be computed. Then in the
174 PRACTICAL ASTRONOMY
triangle ZSM the angle Z and the sides ZS and ZM are known,
because the refraction and parallax corrections are known, and
MS may be computed. By interpolating in the tables, the
true G. M. T. corresponding to the instant of this observation
may be obtained, the difference between this and the observed
chronometer time being the error of the chronometer on G. M.T.
The longitude may then be found by comparing the true G. M. T.
with the local time computed from the sun's altitude.
In the Ephemeris for 1912 the tables of lunar distances have
been omitted, as lunar observations are no longer considered
to be of practical value to the navigator.
109. Azimuth of the Sun at a Given Time.
For determining the error of the compass and for other pur-
poses it is frequently necessary at sea to know the sun's azimuth
at an observed instant of time. If the observed time be con-
verted into local apparent time the azimuth Z may be computed
by the following formulae.*
tan \ (Z+S] = cot \ P sec | (p + co-L) cos | (p co-L) , 
tan \ (Z S) = cot f P esc \ (p-\-co-L) sin | (p co-L). 
In these formulae co-L is the co-latitude. In practice the azimuth
is taken from tables computed by use of these formulae.
Burdwood's and Davis's Azimuth Tables give the azimuth for
each degree of P, L, and p, the former ranging from Lat. 30 to
Lat. 60 and the latter from 30 N to 30 S. Publication
No. 71 of the U. S. Hydrographic Office gives azimuths of the
sun for latitudes up to 61. For finding the azimuth of an
object having a declination greater than 24 publication No. 120
of the Hydrographic Office may be used.
Find the sun's azimuth when L = 42 01' N, D = 22 47' S, P = g h 25 i8 s .
From Publ. No. 71 for L = 42, D = 22, P = q h 20, the azimuth is N 141 40' E.
The corresponding azimuth for L = 43 is 141 50', that is, an increase of 10'
for i; the azimuth for L = 42 D = 23, and P = g h 20, is 142 n', or an
increase of 31' for i of declination; for L = 42, D = 22, and P = g h 30 the
* Napier's Analogies.
azimuth is 143 47', or an increase of 2 07' for io m , or 12'. 7 for i m . The desired
azimuth is therefore 141 40' + & X 10' + f X 31' + 5.3 X i2 ; .7 = 143 12'.
The azimuth from the south point is therefore S 36 48' E.
When the azimuth is determined for the purpose of finding the error of the com-
pass the observation is usually taken near sunrise or sunset, which is not only a
convenient time for making the pointings at the sun but is a favorable time for ac-
curate determination of the azimuth.
no. Azimuth of the Sun by Altitude and Time.
When the altitude of the sun is observed for the purpose of
finding the local time, the azimuth at the same instant may be
computed by the formula
sin Z = sin P cos D sec h. 
Find the sun's azimuth when P= 34 46'.4 (P.M.), D = 22 45' 50", h =
log sin P = 9. 75612
log cos D = g. 96478
log sec h = o. 02 102
log sin Z = g. 74192
Z =S33 3 o'.2W
in. Sumner's Method of Determining a Ship's Position.*
If the declination of the sun and the Greenwich Apparent
Time are known at any instant, these two coordinates are the
latitude and longitude respectively of a point on the earth's
surface which is vertically under the sun's centre and which
may be called the " sub-solar " point. If an observer were at
the sub-solar point he would have the sun in his zenith. If
he were located i from this point, in any direction, the sun's
zenith distance would be i; if he were 2 away, the zenith
distance would be 2. It is evident, then, that if an observer
measures an altitude of the sun he locates himself on the cir-
cumference of a circle whose centre is the sub-solar point and
whose radius (in degrees) is the zenith distance of the sun.
This circle could be drawn on a globe by first plotting the posi-
tion of the sub-solar .point by means of its coordinates, and
* This method was first described by Captain Sumner in 1843.
then setting a pair of dividers to subtend an arc equal to the
zenith distance (by means of a graduated circle on the globe)
and describing a circle about the sub-solar point as a centre.
The observer is somewhere on this circle because all positions
on the earth where the sun has this measured altitude are located
on this same circle. This is called a circle of position, and any
portion of it a line of position or a Sumner line.
Suppose that at Greenwich Apparent Time i h the sun is
observed to have a zenith distance of 20, the declination being
20 N. The sub-solar point is then at A . Fig. 69, and the observer
is somewhere on the circle described about A with a radius 20.
If he waits until the G. A. T. is 4^ and again observes the sun,
obtaining 30 for his zenith distance, he locates himself on the
circle whose centre is B, the coordinates being 4^ and (say)
20 02' N, and the radius of which is 30. If the ship's position
NAUTICAL ASTRONOMY 177
has not changed between the observations it is either at S or
at T; in practice there is no difficulty in deciding which is the
correct point, on account of their great distance apart. A
knowledge of the sun's bearing also shows which portion of the
circle contains the point. If, however, the ship has changed its
position since the first observation, it is necessary to allow for
its " run " as follows. Assuming that the ship has sailed
directly away from the sun, say a distance of 60 miles or i,
then, if the first observation had been made while the ship was
in the second position, the point A would be the same, but the
radius of the circle would be 21, locating the ship on the dotted
circle. The true position of the ship at the second observation
is, therefore, at the intersection S'. If the vessel does not actu-
ally sail directly away from or directly toward the sun it is a
simple matter to calculate the increase or decrease in radius
due to the change in the observer's zenith.
This is in principle Sumner's method of locating a ship.
In practice the circles would seldom have as short radii as those
in Fig. 69; small circles were chosen only for convenience in
illustrating the method. On account of the long radius of the
circle of position only a small portion of this circle can be shown
on an ordinary chart; in fact, the portion which it is necessary
to use is generally so short that the curvature is negligible and
the line of position appears on the chart as a straight line. In
order to plot a Sumner line on the chart, two latitudes may be
assumed between which the actual latitude is supposed to lie;
and from these, the known declination, the observed altitude,
and the chronometer reading, two longitudes may be computed
(Art. 107), one for each of the assumed latitudes. This gives
the coordinates of two points on the line of position by means
of which it may be plotted on the chart. Another observation
may be made a few hours later and the new line plotted in a
similar manner. In order to allow for the change in the radius
of the circle due to the ship's run between observations, it is
only necessary to move the first position line parallel to itself
i 7 8
in the direction of the ship's course and a distance equal to the
ship's run. In Fig. 70, AB is a line obtained from a 9 A.M
observation on the sun and by assuming the latitudes 42 and
43. A second observation is made at 2 P.M.; between 9^ and
2 h the ship has sailed S 75 W, 67'.* Plotting this run on the
chart so as to move any point on AB, such as x, in the direction
S 75 W and a distance of 67', the new position line for the first
observation is A'B'. The P.M. line of position is located by
assuming the same latitudes, 42 and 43, the result being the
line CD. The point of intersection S is the position of the ship
at the time of the second observation. Since the bearing of
the sun is always at right angles to the bearing of the Sumner
line, it is evident that one point and the bearing would be
sufficient to locate the line on the chart.
112. Position by Computation.
The coordinates of the point of intersection of the lines of position may be
calculated more precisely than they can be taken from the chart. When the first
* The nautical mile (6080.27 f eet ) is assumed to be equal to an arc of i' on
any part of the earth's surface.
altitude is measured the navigator assumes a latitude which is near the true lati-
tude, and from this calculates the corresponding longitude. The approximate
azimuth of the sun is also calculated from the same data. (Equa. .) The
run of the ship up to the time of the second observation is reduced to the difference
in latitude and the difference in longitude from the known course and speed of
the vessel. These two differences are applied as corrections to the assumed lati-
tude and the calculated longitude. This places the ship on the new Sumner line
(corresponding to A'B', Fig. 70). When the P.M. observation is made the corrected
latitude is used in computing the new longitude. The result of these two obser-
vations is shown in Fig. 71. Point A is the first position; A' is the position of A
corrected for the run of the ship; B is the position obtained by the P.M. observation
using the latitude of A'. A'B is then the discrepancy in the longitudes, due to
the fact that a wrong latitude has been chosen, and is the base of a triangle the
vertex of which, C, is the true position of the ship. The base angles A' and B
are the azimuths of the sun at the times of observation. In practice this triangle
is often solved as follows:* Dropping a perpendicular from C to A'B, forming
two right triangles,
Bd = Cd cot Z 2 ,
A'd = CdcotZi,
A/> 2 = AL cot 2,
A/>i = AL cot Zi,
* See A. C. Johnson's " On Finding the Latitude and Longitude in Cloudy
180 PRACTICAL ASTRONOMY
where AL = the error in latitude and Ap the difference in departure. In order
to express Bd and A'd as differences in longitude (AM) it is necessary to introduce
the factor sec L, giving
AM 2 = AL sec L cotZ 2 , 
AM 2 = AL sec L cot Z\. 
To find AL, the correction to the latitude, the distance A'B = AM 2 + AMi
is known, the factors sec L cot Z may be found from the approximate latitude
and the sun's azimuths, therefore
sec L cot Zi + sec L cot Z 2
Having found AL, the corrections AMi and AM 2 are found by  and .
Since the factors sec L cot Z are, in practice, taken from a table and the operations
indicated in Equa. , , and  are easily performed with the slide rule
the method is in reality a rapid one.
In the above description the observations are taken one in the forenoon and one
in the afternoon, but any two observations, provided the position lines intersect at an
angle over 30, will give good results. If the observations are both on the same
side of the meridian the denominator of  becomes the difference of the factors
instead of the sum. If two objects can be observed at the same time, and their
bearings differ by 30 or more, the position of the ship is obtained at once, since
there is no run of the ship to be applied. This observation might be made upon
two bright stars or planets at twilight. It should be observed that the accuracy
of this method depends upon the accuracy of the chronometer, just as in the
methods of Art. 107.
One of the great advantages of this method is that even if only one observation
can be taken it may be utilized to locate the ship along a (nearly) straight line;
and this is often of great value. For example, if the first position line is found
to pass directly through some point of danger, then the navigator knows the
bearing of the point, although he does not know his distance from it; but with
the single observation he is able to avoid the danger. In case it is a point which
it is desired to reach, the true course which the ship should steer is at once known .
The following example illustrates the method of computing the coordinates of the
point of intersection.
NAUTICAL ASTRONOMY l8l
Location of ship by Sumner's Method, Jan. 4, 1910.
At chronometer time i h 12 48* the sun's lower limb is observed to be 15 53'
30"; index corr. = o"; height of eye = 36 ft.; chronometer is 15* fast of G. M. T.
Latitude by dead reckoning, 42 oo' N. At chronometer time 6 h 05"* 46 s the alti-
tude of the sun's lower limb = 17 33' 30"; index corr. = o"; height of eye, 36 ft;
chronometer, 15 s fast. The run between the observations was i' N and 60' W.
Semidiam. = + 16' 17" Declination at G. M. N. = 22 47' 22". 3
dip = - 5 53 + i5"-i5 X i h .2 + 18 . 2
r&p = 3 14
Decl. = 22 47' 04". i
Corr. = + 07' 10" p = 112 47' 04". i
Obs. Alt. = 15 53' 30"
True Alt. = 16 oo' 40"
L = 42 oo' seco. 12893 Equa. t. = 4 49*. 80
p = 112 47 csco.03528 i s .i45 X i h .2 = i .37
h = 16 oo . 7
Cor'd Eq. t. = 4"* 51*. 17
170 47 .7
5 = 85 23'. 8 cos 8. 90448 Sun's Az.* = S 36 48' E
s h = 69 23 . i sin 9. 97126 cot Az. X sec Lat. = i. 80
log sin P = g. 51998
% P = 19 20'. 2
P = 38 40'. 4
= 2 h 34 m 41*. 6
L. A. T. = g h 25 i8 s . 4
Eq. t. = 451.2
L. M. T. = g h 30 m 09*. 6
G. M. T. = i 12 33
Long. = 3^ 42 23". 4 Lat. = 42 oo'
= 55 35'- 8 Run = i'
Run = + 60'
Cor'd Lat. = 42 01'
Cor'd Long. = 56 35'. 8
* By table or by Equa. ; see Art. 93, p. 155.
Obs. Alt. =
+ i6' 17" Decl. at G. M. N. =
- 5 53 + I5"i5 X 6*.i =
- 2 54
+ /30" p =
- 22 47' 22". 3
+ i 32 .4
- 22 45' 49". 9
112 45' 49". 9
True Alt. = 17 41' <*>'
L = 42 01' sec o. 1 2904
p = 112 45-8 CSC O.O3522
h= 17 41
"Eq. t. = 4 498. 80
. 145 X 6.i = 6 . 98
Cor'd Eq. t. = 4 56*. 78
172 27 . 8
s = 86 13. 9 cos 8. 81771 Sun's Az.* = S 33 30' W
s h = 68 32'. 9 sin 9. 96883 cot Az. X sec Lat. = 2. 03
log sin P = 9. 47540
I P = 17 23'. 2
L. A. T. = 2* i9 os 8 . 6
Eq. t. = 4 56 . 8
L. M. T. = 2^ 24"* 02*. 4
G. M.T. = 6 05 31
Long. = 3* 4i m 28*. 6
= 55 22'. i
ist Long. = 56 35'. 8 19. 2 X i. 80 = 34'. 6 Corr. to ist Long.
2d Long. = 55 22 . i 19. 2 X 2. 03 = 39 . o Corr. to 2d Long.
Diff. = i 13'. 7 = 73 '. 7 ist Long. = 56 35'. 8 2d Long. = 55 22'.!
Corr. = 34 . 6 Corr. = 39 .o
i. 80 + 2 03 = IQ/ " 2 Corr> to ^ e Lat - Long- = 56 i'- 2 Long. = 56 oi'.i
.'. Lat. = 42 20' N. /. Long. = 56 01' W.
* By table or by Equa. ; see Art. 93, p. 155.
TABLE I. MEAN REFRACTION.
Barometer, 2 9. 5 inches.
Thermometer, 50 P.
2 3 2
2 I 7
5 2 oo
o 3 8
TABLE II. FOR CONVERTING SIDEREAL INTO MEAN SOLAR
(Increase in Sun's Right Ascension in Sidereal h. m. s.)
Mean Time = Sidereal Time C'.
2 5 6 -93 2
3 6 -762
TABLE III. FOR CONVERTING MEAN SOLAR INTO SIDEREAL
(Increase in Sun's Right Ascension in Solar h. m. s.)
Sidereal Time = Mean Time + C.
I 48 . 42 1
PARALLAX SEMIDIAMETER DIP.
(A) Sun's parallax.
(C) Dip of the sea horizon.
Height of eye
Dip of sea
(B) Sun's semidiameter.
1 6' 18"
4 3 6
I 5 47
1 6 09
Local Mean (Astronomical) Times of Culmination and Elongation of Polaris
for 1910, computed for Longitude 90 West of Greenwich and for Latitude 40 N.*
12* 3.9 OT
l& h 42
A 4Q m
Tan. i5. .
1 6 7,9
Feb. i $
Apr. i 5 . .
I 1 ? ^7
Tune i 5 . .
Tuly i . .
Tuly i 5 . .
Aug. i 5 . .
Dec. i5. .
* This table may be used to find the approximate times for any year. For dates
falling between those given in the table the times may be found by interpolation,
the daily difference being about 4. For the method of converting this local time
into Standard time see Art. 35.
TABLE VI. CORRECTION TO THE ALTITUDE OF POLARIS*
(Equa. , Art. 69.)
* This table is calculated for a polar distance = i 10'. An increase of i' in the
polar distance produces an increase of about 3% in the tabulated term. The hour
angle in the table is measured from o at upper culmination either to the east or to
VALUES OF FACTOR 112.5 X 3600 X SIN i" TAN Z e .
3 8 3
Name. Letters. Name.
N, v, Nu
B , 0,
H, f, Xi
r , 7,
O, o, Omicron
n, TT, pi
E , e ,
P, /, Rho
2, cr, 5, Sigma
, r, Tau
T, f, Upsilon
< I ) , </>, Phi
X, x , Chi
, ^, Psi
fi, G>, Omega
ABBREVIATIONS USED IN THIS BOOK
T or V = vernal equinox.
R. A. or Rt. Asc. = right ascension.
D or Decl. = declination.
p = polar distance.
h or Alt. = altitude.
z = zenith distance.
P or H. A. = hour angle.
L or Lat. = latitude.
Long. = longitude.
Sid. = sidereal.
Sol. = solar.
G. M. N. = Greenwich Mean Noon.
G. M. T. = Greenwich Mean Time.
G. A. T. = Greenwich Apparent Time.
G. S. T. = Greenwich Sidereal Time.
L. M. N. = Local Mean Noon.
L. M. T. = Local Mean Time.
L. A. T. = Local Apparent Time.
L. S. T. = Local Sidereal Time.
Eq. T. = equation of time.
Astr. = astronomical time.
Civ. = civil.
E. S. T. = Eastern Standard Time.
U. C. = upper culmination.
L. C. = lower culmination.
or U. L. = upper limb.
or L. L. = lower limb.
RL ,LL = right limb, left limb.
* = star.
Corr. = correction.
I. C. = index correction,
r. or refr. = refraction correction.
p = parallax correction.
s. d. = semidiameter.
Z or Az. = azimuth.
N, E, S, W = north, east, south, west.
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 periodic rise and fall of the surface of the ocean, caused
by the moon's and the sun's attraction, is called the tide. The
word " tide " is sometimes applied to the horizontal movement
of the water (tidal currents), but in the following discussion
it will be used only to designate the vertical movement. When
the water is rising it is called flood tide; when it is falling it is
called ebb tide. The maximum height is called high water; the
minimum is called low water. The difference between the two
is called the range of tide.
Cause of the Tides.
The principal cause of the tide is the difference in the force
of attraction exerted by the moon upon different parts of the
earth. Since the force of attraction varies inversely as the
square of the distance, the portion of the earth's surface nearest
the moon is attracted with a greater force than the central
portion, and the latter is attracted more powerfully than the
portion farthest from the moon. If the earth and moon were
at rest the surface of the water beneath the moon would be
THE TIDES 193
elevated as shown in Fig. 72 at A. And since the attraction
at B is the least, the water surface will also be elevated at this
point. The same forces which tend to elevate the surface at
A and B tend to depress it at C and D. If the earth were
set rotating, an observer at any point O, Fig. 72, would be
carried through two high and two low tides each day, the approx-
imate interval between the high and the low tides being about
6j hours. This explanation shows what would happen if
the tide were developed while the two bodies were at rest; but,
owing to the high velocity of the earth's rotation, the shallow-
ness of the water, and the interference of continents, the actual
tide is very complex. If the earth's surface were covered with
water, and the earth were at rest, the water surface at high
tide would be about two feet above the surface at low tide.
The interference of continents, however, sometimes forces the
tidal wave into a narrow, or shallow, channel, producing a
range of tide of fifty feet or more, as in the Bay of Fundy.
The sun's attraction also produces a tide like the moon's,
but considerably smaller. The sun's mass is much greater
than the moon's but on account of its greater distance the ratio
of the tide-producing forces is only about 2 to 5. The tide
actually observed, then, is a combination of the sun's and the
Effect of the Moon's Phase.
When the moon and the sun are acting along the same line, at
new or full moon, the tides are higher than usual and are called
spring tides. When the moon is at quadrature (first or last quar-
ter), the sun's and the moon's tides partially neutralize each other
and the range of tide is less than usual ; these are called neap tides.
Effect of Change in Moon's Declination.
When the moon is on the equator the two successive high
tides are of nearly the same height. When the moon is north
or south of the equator the two differ in height, as is shown in
Fig. 73. At. point B under the moon it is high water, and the
depth is greater than the average. At B', where it will again
be high water about 12 later, the depth is less than the average.
This is known as the diurnal inequality. At the points E and Q,
on the equator, the two tides are equal.
Effect of the Moon's Change in Distance.
On account of the large eccentricity of the moon's orbit
the tide-raising force varies considerably during the month.
The actual distance of the moon varies about 13 per cent, and
as a result the tides are about 20 per cent greater when the moon
is nearest the earth, at perigee, than they are when the moon
is farthest, at apogee.
THE TIDES 195
Priming and Lagging of the Tides.
On the days of new and full moon the high tide at any place
follows the moon's meridian passage by a certain interval of
time, depending upon the place, which is called the establish-
ment of the port. For a few days after new or full moon the
crest of the combined tidal wave is west of the moon's tide and
high water occurs earlier than usual. This is called the priming
of the tide. For a few days before new or full moon the crest
is east of the moon's tide and the time of high water is delayed.
This is called lagging of the tide.
All of these variations are shown in Fig. 74, which was con-
. structed by plotting the predicted times and heights from the U. S.
Coast Survey Tide Tables and joining these points by straight
lines. It will be seen that at the time of new and full moon the
range of tide is greater than at the first and last quarters; at the
points where the moon is farthest north or south of the equator
(shown by N, S,) the diurnal inequality is quite marked,
whereas at the points where the moon is on the equator ()
there is no inequality; at perigee (P) the range is much greater
than at apogee (A).
Effect of Wind and Atmospheric Pressure.
The actual height and time of a high tide may differ consider-
ably from the normal values at any place, owing to the weather
conditions. If the barometric pressure is great the surface is
depressed, and vice versa. When the wind blows steadily into
a bay or harbor the water is piled up and the height of the tide
is increased. The time of high water is delayed because the
water continues to flow in after the true time of high water has
passed; the maximum does not occur until the ebb and the effect
of wind are balanced.
Observation of the Tides.
In order to determine the elevation of mean sea level, or,
more properly speaking, of mean half-tide, it is only necessary
to observe, by means of a graduated staff, the height of high
and low water for a number of days, the number depending upon
THE TIDES 197
the accuracy desired, and to take the mean of the gauge read-
ings. If the height of the zero point of the scale is referred to
some bench mark, by means of a line of levels, the height of the
bench mark above mean sea level may be computed. In order
to take into account all of the small variations in the tides
it would be necessary to carry on the observations for a series
of years; a very fair approximation may be obtained, however,
in one lunar month, and a rough result, close enough for many
purposes, may be obtained in a few days.
If an elaborate series of observations is to be made, the self-
registering tide gauge is the best one to use. This consists of
a float, which is enclosed in a vertical wooden box and which
rises and falls with the tide. A cord is attached to the float
and is connected by means of a reducing mechanism with the
pen of a recording apparatus. The record sheet is wrapped
about a cylinder, which is revolved by means of clockwork.
As the tide rises and falls the float rises and falls in the box
and the pen traces out the tide curve on a reduced scale. The
scale of heights is found by taking occasional readings on a
staff gauge which is set up near the float box and referred to a
permanent bench mark. The time scale is found by means of
reference marks made on the sheet at known times.
When only a few observations are to be made the staff gauge
is the simplest to construct and to use. It consists of a vertical
graduated staff fastened securely in place, and at such a height
that the elevation of the water surface may be read on the
graduated scale at any time. Where the water is compara-
tively still the height may be read directly on the scale; but
where there are currents or waves the construction must be
modified. If a current is running rapidly by the gauge but
the surface does not fluctuate rapidly, the ripple caused by the
water striking the gauge may be avoided by fastening wooden
strips on the sides so as to deflect the current at a slight
angle. The horizontal cross section of such a gauge is shown in
Fig. 75. If there are waves on the surface of the water the height
will vary so rapidly that accurate readings cannot be made. In
order to avoid this difficulty a
glass tube about f inch in di-
ameter is placed between two
wooden strips (Fig. 76), one of
which is used for the graduated
scale. The water enters the glass tube and stands at the height
of the water surface outside. In order to check sudden varia-
tions in height the water is allowed to enter this tube only
through a very small tube (i mm inside diameter) placed in a
cork or rubber stopper at the lower end
of the large tube. The water can rise
in the tube rapidly enough to show the
general level of the water surface, but
small waves have practically no effect
upon the reading. For convenience the
gauge is made in sections about three
feet long. These may be placed end to
end and the large tubes connected by
means of the smaller ones passing
through the stoppers. In order to read
the ' gauge at a distance it is convenient
to have a narrow strip of red painted
on the back of the tube or else blown
into the glass.* Above the water surface
this strip shows its true size, but below
the surface, owing to the refraction of
light by the water, the strip appears
several times its true width, making
it easy to distinguish the dividing line.
Such a gauge may be read from a considerable distance by
means of a transit telescope or field glasses.
* Tubes of this sort are manufactured for use in water gauges of steam boilers.
THE TIDES 199
Location of Gauge.
The spot chosen for setting up the gauge should be near the
open sea, where the true range of tide will be obtained. It
should be somewhat sheltered, if possible, against heavy seas.
The depth of the water and the position of the gauge should be
such that even at extremely low or extremely high tides the
water will stand at some height on the scale.
Making the Observations.
The maximum and minimum scale readings at the times of
high and low tides should be observed, together with the times
at which they occur. The observations of scale readings should
be begun some thirty minutes before the predicted time of high
or low water, and continued, at intervals of about 5 m , until a
little while after the maximum or minimum is reached. The
height of the water surface sometimes fluctuates at the time
of high or low tide, so that the first maximum or minimum
reached may not be the true time of high or low water. In
order to determine whether the tides are normal the force and
direction of the wind and the barometric pressure may be
Reducing the Observations.
If the gauge readings vary so that it is difficult to determine
by inspection where the maximum or minimum occurred, the
observations may be plotted, taking the times as abscissae and
gauge readings as ordinates. A smooth curve drawn through
the points so as to eliminate accidental errors will show the posi-
tion of the maximum or minimum point. (Figs, yya and yyb.)
When all of the observations have been worked up in this way
the mean of all of the high-water and low-water readings may
be taken as the scale reading for mean half-tide. There should
of course be as many high-water readings as low- water readings.
If the mean half- tide must be determined from a very limited
number of observations, these should be combined in pairs
in such a way that the diurnal inequality does not introduce
an error. In Fig. 78 it will be seen that the mean of a and b,
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
MACHIAS BAY, ME.
JUNE 8, 1905.
one case too small and in the other case too great. The proper
selection of tides may be made by examining the predicted
heights and times given in the tables issued by the U. S. Coast
THE TIDES 201
and Geodetic Survey. By examining the predicted heights the
exact relation may be found between mean sea level and the
mean half-tide as computed from the predicted heights corre-
sponding to those tides actually observed. The difference be-
tween these two may be applied as a correction to the mean
of the observed tides to obtain mean sea level. For example,
suppose that the predicted heights at a port near the place of
observation indicate that the mean of a, b, c, d, e, and/ is 0.2 ft.
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
annually by the United States Coast and Geodetic Survey;
these tables give the times and heights of high and low water
for the principal ports of the United States, and also for many
foreign ports. The method of using these tables is explained
in a note at the foot of each page. A brief statement of the
theory of tides is given in the Introduction.
. The approximate time of high water at any place may be
computed from the time of the moon's meridian passage, pro-
vided we know the average interval between the moon's transit
and the following high water, i.e., the " establishment of the
port." The mean time of the moon's transit over the meridian
of Greenwich is given in the Nautical Almanac for each day,
together with the change per hour of longitude. The local
time of transit is computed by adding to the tabular time the
hourly change multiplied by the number of hours in the west
longitude; this result, added to the establishment of the port,
gives the approximate time of high water. The result is nearly
correct at the times of new and full moon, but at other times
is subject to a few minutes variation.
Aberration of light, 12
Adjustment of transit, 83, 87
Almucantar, 15, 83, in
of pole, 27
Angle of the vertical, 73
Annual aberration, 13
Apparent motion, 3, 28
Arctic circle, 30
Aries, first point of, 16
Astronomical time, 44
transit, 87, 117
triangle, 31, 120
Atlantic time, 56
Attachments to transit, 86
Autumnal equinox, 16
Axis, 3, 8
Azimuth, 19, 146
Besselian year, 68
Celestial latitude and longitude, 22
Central time, 56
Chronograph, 93, 136, 141
Chronometer, 92, 141, 173
Circumpolar star, 29, 103, 155
Civil time, 44
Comparison of chronometer, 93
Constant of aberration, 13
Constellations, 10, 98
Cross hairs, 82, 87
Culmination, 39, 61, 103
Curvature, 120, 153, 158
Date line, 58
Dead reckoning, 172
parallels of, 16
Diurnal aberration, 13, 158
Eastern time, 56
Ebb tide, 192
Ecliptic, 16, loo, 102
Elongation, 36, 147
Equal altitude method, 128, 164
Equation of time, 41
Equinoxes, 9, 16
Errors in horizontal angle, 97
in transit observations, 88, 118
Eye and ear method, 96
Eyepiece, prismatic, 87, 118
Figure of the earth, 72
Fixed stars, 2, 4, 68
Flood tide, 192
Greenwich, 23, 45, 52, 172
system, 19, 83
Hour angle, 20, 36
Hydrographic office, 134, 174
Index error, 84, 90, 106
Latitude, 22, 27, 103
astronomical, geocentric and geodetic,
at sea, 170
reduction of, 73
Level correction, 158
Local time, 45
Longitude, 22, 45, 139
at sea, 172
Lunar distance, 172
Mean sun, 41, 55
Micrometer, 94, 112, 156
Midnight sun, 30
Moon, apparent motion of, 5
culminations, 69, 141
Motion, apparent, 3, 28
Mountain time, 56
Nautical almanac, 43, 62
Neap tide, 194
Object glass, 82, 87
Obliquity of ecliptic, 8, n, 16
Observer, coordinates of, 22
of earth, 7
Pacific time, 56
Parallactic angle, 31, 134, 154
Parallax, 63, 73
Parallel of altitude, 15
of declination, 16
Phases of the moon, 144, 194
Planets, 3, 102, 135
Plumb-line, 14, 72
Pole, 3, ii, 15
star, 99, 162
Polar distance, 20
Precession, 10, 101
Prediction of tides, 201
Primary circle, 18
Prime vertical, 16, 122, 172
Prismatic eyepiece, 87, 118, 152
Radius vector, 41
of tide, 192
Reduction to elongation, 150
of latitude, 73
to the meridian, 109
effect on dip, 80
index of, 77
Retrograde motion, 6
Right ascension, 20, 36
Rotation, 3, 39
Run of ship, 177
Secondary circles, 18
Semidiameter, 63, 78
contraction of, 79
Sextant, 80, 88, 170
Sidereal day, 39
time, 40, 49, 52
Signs of the Zodiac, 100
Solar day, 40
time, 40, 49, 52
Solstice, 1 6
Spherical coordinates, 18, 31
Spheroid, 10, 72
Spirit level, 14
Spring tides, 194
Stadia hairs, 151
Standard time, 56
Standards of transit, 82
Star catalogues, 69, 94
list, 119, 130
Striding level, 86, 87, 115, 156
Sub-solar point, 175
Sumner's method, 175
Sumner line, 176, 179
Sun, altitude of, 105
apparent motion of, 5
Talcott's method, 69, 94, 112
Telegraph method, 140
Tide gauge, 197
Time ball, 137
Transit, astronomical, 87
engineer's, 78, 82
time of, 39
Transportation of timepiece, 139
Tropical year, 50
Vernal equinox, 16
Vernier of sextant, 90
of transit, 82
Vertical circle, 14, 124
Visible horizon, 14
Watch correction, 114, 139
Wireless telegraph signals, 137
Year, 50, 68
telescope, 94, 112
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