= ^ {h + s^sin^a.C f /i.s^sin aE),
or with ample precision
S4> (for 15 miles or less) = — {h + s^ sin^ aC).
In either case
and
and
^'  (f> + Jcj) = latitude of second station;
V cos /
i^' = ^ + J /< = longitude of second station ;
or with ample precision
'ia (for 15 miles or less) = — (/O sin i{'),
COMPUTING THE GEODETIC POSITIONS 115
which agrees with the result of Art. 70. The sign of ^a is for
the northern hemisphere, and is to be reversed in the southern
hemisphere. Then
«' = c: + Jn: + 180° = back azimuth at second station.
In the above formulas the values of ^i^, '^^, and ^a are obtained
in seconds. In using the formulas both north and south latitude
are to be taken as positive, west longitude as positive and east
longitude as negative, and the trigonometric functions are to be
given their proper signs. The lettered factors of the formulas
have the following values :
A = A'il  e^ sin2 ^')i, D = D'(f^^),
^ \1— e^ sm^ (f>/
B =£'(le2sin2 96)?, E =E'{1 + 3 tan^ 4>) (1  e^ sin^
, F = F' (sin cos^ 4>),
G = value determined by second part of Table II,
in which the logarithms of the constants are as follows:
Metric. Feet.
8.5097218  10 7.9937376  10
8.5126714  10 7.9966872  10
1.4069381  10 0.3749697  10
2.6921687  20 2.6921687  20
5.6124421  20 4.5804737  20
8.2919684  20 8.2919684  20
With the help of these constants it is not difficult to find
the values of the factors A to F for any latitude. If the distance
s is given in meters these factors may be taken from Table II,
at the end of the book, this table being an abridgment of the
Constant.
A'
1
a arc 1"
B'
1
a(l  e2) arc 1"
C
1
2a2(le2)arcl"
D'
= e2arcl"
E'
1
6a2
F'
= ±arc2 1"
116 GEODETIC SURVEYING
Coast Survey tables referred to (and corrected to agree with
the U. S. legal meter of 39.37 inches).
72. The Clarke Solution. This solution of the problem (Art.
64) is adapted to greater distances than the previous one, being
sufficiently precise for the longest lines (say about 300 miles) that
could ever be directly observed. It has the advantage of being
reasonably convenient in use, even without specially prepared
tables, but requires not less than nine place logarithms for close
work, on account of the size of the numbers involved. In this
method the azimuthal angles are used in the computations
instead of the azimuths themselves. The azimuthal angles
(shown in Fig. 34, page 108, and explained at end of Art. 68),
are the angles (at the stations) inside the polar triangles which
are formed by the nearest pole and the two stations, the relation
to the corresponding azimuths being always selfevident. The
formulas used in this solution (taken from Appendix No. 9,
Report for 1885, U. S. Coast and Geodetic Survey, but modi
fied in form) are as follows:
Let 4> = the known latitude at the first station;
X = the known longitude at the first station;
a = the known azimuthal angle at the first station;
j)' = the unknown latitude at the second station ;
X' = the unknown longitude at the second station;
a' = the imknown azimuthal angle at the second station;
s = the known distance between the stations;
6 = the angle between terminal normals;
X, = auxiliary azimuthal angle at second station;
AX = X' — k = difference of longitude;
J
+ (j)') = middle latitude.
From Art. 69,
iV= " R  "(1 «^)
(1  e2 sin2
+ ')]r
Then
^ = M ^ Ml + ( Lf"^ 2\ r^ ^'^^^ 9^ ^=082 a.
N sm 1 \ 6(1 — e'') /
COMPUTING THE GEODETIC POSITIONS 117
But if s is not over about 100 miles we may write with ample
precision
N sin 1""
In either case s and N must be in the same unit, and 6 is obtained
in seconds. If the second term is used in finding d the approx
imate value of 6 is used in that term. The value of this second
term is always extremely small. Then
p Sin 1
' 6^ 008^ (p sin 2a,
4(1 e2)
in which ^^ is obtained in seconds and is always a very small
quantity;
tan P=^^^! cot J
, „ cosKr^) .a
tan Q = . / , „, cot ^,
cos^ (7 + d) 2'
from which values
a' = P + Q — ^ = azimuthal angle at second station;
Ak=Q  P;
X' = X + AX = longitude at second station.
The difference of latitude is found from the formula
f'sin i{a' + ^— a)'
J^ =
i?sinl"Vsini(a:'+ ^ + a)
/sinM:
^\ 12
)^2cos2J(a'a)l
in which ^^ is obtained in seconds, and in which s and R must
be in the same unit. Then
must therefore be found by successive approximation
— that is, an approximate value of R must first be used to obtain
an approximate value of ^0, a greatly improved value of R thus
becoming available to find a much closer value of 4^, and so on.
118
GEODETIC SURVEYING
A few trials will soon give a value of R which is consistent with the
value of )^ + E{A<}>)y^ + ECy%
from which we obtain
tan a = — and
X
y
X
sm a cos a
COMPUTING THE GEODETIC POSITIONS
119
The closest value of s is obtained from the fraction whose numer
ator is the smallest. Then, from Art. 71,
1
za =  r
{JX)smi(+4>')
{AX)^F
cosi(J^)
Aa (for 15 miles or less) = —{AX) sin i(^+^');
and in either casi^
a' = a + Aa + 180°
Either station may be called the first station, so that the problem
may be worked both ways as a check, if desired, in which case
Aa need not be computed at all. As in Art. 71, the values
Aa^ J(j)^ and AX are expressed in seconds, and s will be in the
same xmit as that on which the factors A, B, etc., are based.
By the Clarke Formulas. In this method the desired values
are found by successive approximation. The Puissant method
Fig. 37.
is applied first, therefore, to obtain as close an approximation
as possible to begin with. The approximate values of s and a
(changed to the azimuthal angle) are then substituted in the
Clarke formulas, calling either station the first station, and com
puting the latitude and longitude for the second station. The
computed values will usually disagree a small amount with the
known latitude and longitude of the second station, and a new
trial has to be made with s and a slightly changed, and so on
until the assimied values of s and a satisfy the known con
ditions. The disagreement to be adjusted is always very small,
and when all the circumstances are known it is not difficult to
120 GEODETIC SUEVEYINQ
judge which way and how much to modify s and a to remove
the difficulty. Referring to Fig. 37, let the lines NS represent
meridians, the line CB a parallel of latitude, and A and B the
points whose latitude and longitude are known. With the assumed
distance s and the assumed azimuthal angle a suppose, for
instance, that the computation gives us the point B' instead
of the desired point B. We then have
BC = error in longitude in seconds of arc;
B'C = error in latitude in seconds of arc ;
BB' (in seconds) = V BC^ + B'C^;
b = BB' in distance = {BB')R sin 1" (approximately) ;
tanC5'5 = g;
BB'D = 180°  a'  CB'B;
b cos BB'D = B'D = approximate error in the assumed value
for distance s;
: — 71— = BAD (nearly) in seconds = approximate error in
s sm 1"
assumed value of angle a.
74. Locating a Parallel of Latitude. For marking bound
aries, or other purposes, it often becomes desirable to stake out
a parallel of latitude directly on the ground. Points on the
parallel are most conveniently found by offsets from a tangent
(Art. 69). Thus in Fig. 38, ABD is a tangent from the point
A, and ACF is the corresponding parallel; the point C on the
parallel, for instance, is determined by the offset BC and the
backazimuth angle SB A. It is seldom desirable to run a tangent
over 50 miles on account of the long offsets required ; if the parallel
is of greater length it is better to start new tangents occasionally.
The computations may be made by either the Puissant (Art.
71), or tlie Clarke (Art. 72) formulas, which are much simplified
by the east and west azimuths. Using the Puissant formulas,
substituting 90° (westward) or 270° (eastward) for a, and omitting
COMPUTING THE GEODETIC POSITIONS 121
inappreciable terms, we have with great precision for a hundred
miles or more
J4> (in seconds) = — s'^C,
n (in seconds) = f '^''''^^ ^ + 1 ^77 ;
[ running E, — J cos (in linear units) = — (s^C) R sin 1" = om >
in which either formula may be used as preferred, and in 'which
all linear quantities must be taken in the same unit. The
expressions for N and R are given in Art. 69. For the change
of azimuth we have
Ja (in seconds) =
■^;'^"'!f'^";;}[(i^)sini(^+^')+(i^)^i^];
or for the field work (within onetenth of a second),
J r JN f N. hemisphere, .,,^ . ,,
Aa (m seconcjs) = \a u 1  ('^'^) ^^"^ ^9
It is seen from the above formulas that the offsets (in seconds
or linear units) may be taken to vary directly as the square of
the distance, and the change of azimuth directly as the change
of longitude.
In actual practice the point A may have to be located, or
may be given by description or monument; in either case the
latitude and meridian at A are determined by astronomical
122 GEODETIC SURVEYING
observations, and the tangent AB (or a line parallel thereto)
run out by the ordinary method of double centering. At the
end of the tangent the computed value of the back azimuth
should be compared with an astronomical determination; in
the writer's experience on the Mexican Boundary Survey with
an 8inch repeating instrument (with striding level), and heliotrope
sights ranging in length from 6 to 80 miles, the backazimuth
error was readily kept below onetenth of a second per mile,
regardless of the number of prolongations in the line. The
conditions met with in the survey referred to are illustrated
in Fig. 39, which shows also the adjustment made for back
azimuth error. The boundary line was intended to be the parallel
of 31° 47', but according to treaty all existing monuments had
to be accepted as marking the true line. The astronomical
station was conveniently located, and proved to be slightly south
of the desired parallel, which in turn passed south of the old
monument L. When the last point on the tangent was reached
the back azimuth measured less than the theoretical value,
indicating that the tangent as staked out swerved slightly to
the south from its original direction. Assuming all corresponding
distances on tangents and parallels to be equal and the azimuth
error to accumulate uniformly from A to d,
Let E = azimuth error at d;
Eh — azimuth error at &;
then
Eb = ^,E; dD = ^E sinl"; bB = i^Esml
If .
Ad ' 2 ' 2Ad
DF = A(j> (linear) for AD; BC = J
00
»o
o
05
00
CO d
tH
U5
t^
GO
M
05 Q Ol
Oi O OS
CO CO to
to CO (M
CO IC rH
'os 00 I>
a
o o o
O) O^ C3
OS OS OS
CM 0» "O
rH r—l
rt
r^
"5
Hfl
I^ l> 00
1
^ 1
111
M(M cq
U3 00 ^
O CO (N
OS lO o
00 00 QO
^ ^ (N
H O O
lO O lO
<© I> 00
M ?§
!> CO
•^ CO
P=55
>^
<§H
cs
(M
Oi
lO
3 03
c^
00
3
"2 a
£g
G^ G^ 00
to ^ O
00 OS t^
l> ir Tt<
CO l> CO
O 03 C3
OOrH
03 OS OS
Oi Oi 1
« a
M
p
3
u
a>
a
fl
t3
(U
t.
0}
p
s
§
ai
■a
w
a
ta
m
t4
s
1
a
1
S
s
1
o
h
B
'3
CO
o
c
1
^
o
•a
"d
si
3
IS
i
«w
t
■o
1 1 III
s
C!J
3
n
a
u
C5
1
4: III
o
iH
1
i
i_J
o
Fig. 45. — ^Molitor's Preciselevel Rod and Johnson's Footpin.
IGO GEODETIC SUEVETING
best made of steel. The little groove in the head is to prevent
dust or sand from settling on the bearing point.
92. Adjustment of Level Work. In running level lines of
any importance the work is always arranged so as to furnish
a check on itself, or to connect with other systems, and a cor
responding adjustment is required to eliminate the discrepancies
which appear. The problem may always be solved by the method
of least squares when definite weights have been assigned to the
various lines. When the work is all of the same grade the lines
are weighted inversely as their length. This rule requires an
error to be distributed uniformly along any given line to adjust
the intermediate points. A common rule for intermediate points
on a line or circuit is to distribute the error as the square root
of the various lengths; but as this rule is inconsistent with itself
it is not recommended. The following rules for the adjustment
of level work will usually be found sufficient and satisfactory.
Duplicate lines. A duplicate line is understood to mean a
line run over the same route, but in the opposite direction and
with different turning points. This is the best way of checking
a single line of levels. The discrepancy which usually appears
is divided equally between the two lines.
Simultaneous lines. These are lines run over the same route
in the same direction, but with different turning points. In
this case the final elevation is taken as the mean of the elevations
given by the different lines.
Multiple lines. This is understood to mean two or more
lines run between two points by different routes. In this case
the difference of elevation as given by each line is weighted inversely
as the length of that line, and the weighted arithmetic mean
is taken as the most probable difference of elevation. Thus if
the difference of elevation between A and B is 9.811 by a 6mile
line, 9.802 by an 8mile line, and 9.840 by a 12mile line, we have
Mean difference of elevation
_ (9811 X i) + (9.802 X i) + (9.840 X jS)
+ J
= 9.S14.
T2
Intermediate points. These may occur on a line whose ends
have been satisfactorily adjusted or on a closed circuit. In
either case the required adjustment is distributed uniformly
throughout the line, making the correction between any two
GEODETIC LEVELING
161
points directly proportional to the length between those two
points.
Level nets. Any combination of level lines forming a series
of closed circuits is called a polygonal system or level net. Fig. 46
represents such a system. If the true difference of elevation
were known from point to point, then the algebraic sum of the
differences in any closed circuit would always equal zero, the
rise and fall balancing. In practical work the various circuits
seldom add up to zero, and an adjustment has to be made to
eliminate the discrepancies. A rigor
ous adjustment requires the use of
the method of least squares, but the
approximate adjustment here described
will generally give very nearly the same
results. Pick out the circuit which
shows the largest discrepancy, and
distribute the error among the differ
ent lines in direct proportion to their
length. Take the circuit showing the
next largest discrepancy, and distribute
its error imiformly among any of its
lines not previously adjusted in some
other circuit, continuing in this way
until all the circuits have been ad
justed. The circuits here intended are
the single closed figures, as BEFC, and not such a circuit as
ABEFCA; and no attention is to be paid to the direction or
combination in which the lines may have been run.
93. Accuracy of Precise Spirit Leveling. The accuracy
attainable in precise spirit leveling may be judged by noting the
discrepancies between duplicate lines (Art. 92). On the U. S.
Coast and Geodetic Survey the limit of discrepancy allowed
between duplicate lines is 4mm. vX, meaning 4 millimeters
multiplied by the square root of the distance in kilometers between
the ends of the lines; if this limit is exceeded the line must be
rerun both ways until two results are obtained which fall within
the specified limits. In various important surveys the allowable
limit has ranged from 5mm. v'i? to 10mm. Vif, or 0.021ft. ^/M
to 0.042ft. Vikf where M is the distance in miles. The probable
error of the mean result of a pair of duplicate lines is practically
162 GEODETIC SUEVEYING
onethird of the discrepancy, and in actual work of the highest
grade falls below Imm.Vif. The adjusted value of the eleva
tion above mean sea level of Coast Survey bench mark K in
St. Louis has a probable error of only 32 millimeters or about
li inches, and it is almost certain that no amount of leveling
will ever change the adopted elevation as much as 6 inches.
A much more severe test of the accuracy of leveling is obtained
from the closures of large circuits running up sometimes to 1000
or more miles in circumference. The greatest error indicated
by the circuit closures in any line in about 20,000 miles of precise
spirit leveling executed by the U. S. Coast and Geodetic Survey
and other organizations, is about onetenth of an inch per mile.
With the Coast Survey level of Art. 90 very much closer results
have been reached.
CHAPTER VII
ASTRONOMICAL DETERMINATIONS
94. General Considerations. The astronomical determina
tions required in practical geodesy are Time, Latitude, Longitude
and Azimuth. The precise determination of these quantities
requires special instruments as well as special knowledge and skill,
and falls within the province of the astronomer or professional
geodesist rather than that of the civil engineer. A fair deter
mination, however, of one or more of these quantities is not
infrequently required of the engineer, so that a partial knowledge
of the subject is necessary. A complete discussion of the sub
jects of this chapter may be found in Doolittle's Practical Astron
omy, or in Appendix No. 7, Report for 189798, U. S. Coast and
Geodetic Survey. As the work of the fixed observatory is out
side the sphere of the engineer, the following articles are intended
to cover field methods only.
The instruments used by the engineer will generally be limited
to the sextant, the engineer's transit, one of the higher grades
of transits, or the altazimuth instruments of Chapter III. All
of these instruments are suitable for either day or night observa
tions, except that the ordinary engineer's transit is not usually
fiurnished with means for illuminating the crosshairs at night.
This difficulty may be overcome by substituting in place of the
sunshade a similar shade of thin white paper, a flat piece of bright
tin bent over in front of the object glass at an angle of about 45°
and containing an oblong hole having a slightly less area than
that of the lens, or a special reflecting shade which may be bought
from the maker of the instrument. The light of a bull'seye
lantern thrown on any of these devices will render the crosshairs
visible.
In astronomical work the observer is assumed to be at the
center of the earth, this point being taken as the center of a great
163
164 GEODETIC SUEVEYING
celestial sphere on which all the heavenly bodies are regarded as
being projected. Any appreciable errors arising from the assumption
that the earth is stationary or that the observer is at its center,
are duly corrected. All vertical and horizontal planes and the
planes of the earth's equator and meridians are imagined extended
to an intersection with the celestial sphere, and are correspond
ingly named. Fig. 47, page 166, is a diagram of the celestial
sphere, and the accompanying text contains the definitions and
notation used in the discussions. A thorough study and compre
hension of the figure and text are absolutely essential for an
understanding of what follows. The necessary values of the
right ascensions, declinations, etc., required in the formulas, are
obtained from the American Ephemeris, commonly called the
Nautical Almanac, which is issued yearly (three years in advance)
by the Government.
Time
95. General Principles. Time is measured by the rotation
of the earth on its axis, which may be considered perfectly uniform
for the closest work. The rotation is marked by the observer's
meridian sweeping around the heavens. The intersection of
this meridian with the celestial equator furnishes a point whose
uniform movement aroimd the equator marks off time in angular
value. The angle thus measured at any moment between the
observer's meridian and the meridian of any given point (which
may itself be moving) is the hour angle of that point at that
moment. These angles are, of course, identical with the cor
responding spherical angles at the pole. When 360° of the equa
tor have passed by the meridian of a reference point (whether
moving or not) the elapsed time is called twentyfour hours, so
that any kind of time is changed from angular value to the hoiu
system by dividing by 15, and vice versa. There are two kinds
of time in common use, mean solar time and sidereal time, based
on the character of the reference point. Mean solar time is the
ordinary time of civil life, and sidereal time is the time chiefly used
in astronomical work.
96. Mean Solar Time. The fundamental idea of solar time is to
use as the measure of time the apparent daily motion of the sun
ASTRONOMICAL DETEEMINATIONS 166
around the earth; this is called apparent solar time, the upper transit
of the sun at the observer's meridian being called apparent noon.
Apparent solar time, however, is not uniform, on account of a
lack of uniformity in the apparent annual motion of the sun
around the earth. This is due to the fact that the apparent
annual motion is in the ecliptic, the plane of which makes an angle
with the plane of the equator, and the further fact that even in
the ecliptic the apparent motion is not uniform. To overcome
this difficulty, a fictitious sun, called the mean sun, is assumed to
move annually around the equator at a perfectly uniform rate,
and to make the circuit of the equator in the same total time that
the true sun apparently makes the circuit of the ecliptic. Mean
solar time is time as indicated by the apparent daily motion of
the mean sun and is perfectly uniform. The difference between
apparent solar time and mean solar time is called the equation
of time, varies both ways from zero to about seventeen minutes,
and is given in the Nautical Almanac for each day of the year.
Local mean time for any meridian is the hour angle of the mean
sun measured westward from that meridian, local mean noon
being the time of the upper transit of the mean sun for that
meridian.
96a. Standard Time. This time, as now used in the United
States, is mean solar time for certain specified meridians, each
district using the time of one of these standard meridians instead
of its own local time. The meridians used are the 75th, 90th,
105th and 120th west of Greenwich, furnishing respectively
Eastern, Central, Mountain and Pacific standard time. Standard
time for all points in the United States differs only by even hours,
with very large belts having exactly the same time, the variation
from local mean time seldom exceeding a half hour. In the lat
itude of New York local mean time varies about four seconds
for every mile east or west. Standard time may be obtained at
any telegraph station with a probable error of less than a second.
In all astronomical work standard time must be changed to local
mean time.
96b. To Change Standard Time to Local Mean Time and vice
versa. The difference between standard time and local mean
time at any point equals the difference of longitude (expresed
in time units. Art. 113) between the given point and the standard
time meridian used. For points east of the standard time
166
GEODETIC SURVEYING
North
Fig. 47.— The Celestial Sphere.
EXPLANATION
^2^^ = meridian of observer;
Z, IF, Af =points on prime vertical;
M, TO = projection of azimuth marks on celestial sphere;
Z = observer's zenith;
A'' = observer's nadir;
Angles at Z, and corresponding horizontal angles at 0, are azimuth angles;
Angels at P, and corresponding equatorial angles at 0, are hour angles.
Conversion op Arc and Time
Arc. Time.
1° = 4 minutes
1' =4 seconds
1" = tV second
Time. Arc.
1 hour = 15°
1 minute = 15'
1 second = 15"
ASTRONOMICAL DETERMINATIONS 167
DEFINITIONS
The zenith (at a given station) is the intersection of a vertical line with
the upper portion of the celestial sphere.
The nadir is the intersection of a vertical line with the lower portion
of the celestial sphere.
The meridian plane is the vertical plane through the zenith and the celes
tial poles, the meridian being the intersection of this plane with the celestial
sphere.
The prime vertical is the vertical plane (at the point of observation) at
right angles with the meridian plane.
The latitude of a station is the angular distance of the zenith from the
equator, and has the same value as the altitude of the elevated pole. Lati
tude may also be defined as the declination of the zenith. North latitude
is positive and south latitude negative.
CoZaiiSude = 90°— latitude.
Right ascension is the equatorial angular distance of a heavenly body
measured eastward from the vernal equinox.
Declination is the angular distance of a heavenly body from the equator.
North declination is positive and south dechnation negative.
Codeclination or polar distance = 90°— declination.
The hour angle of a heavenly body is its equatorial angular distance
from the meridian. Hour angles measured towards the west are positive,
and vice versa.
The azimuth of a heavenly body (or other point) is its horizontal angular
distance from the south point of the meridian (unless specified as from the
north point). Azimuth is positive when measured clockwise, and vice
versa.
The altitude of a heavenly body is its angular distance above the horizon.
Coaltitude or zenith distoKce = 90°— altitude.
Refraction is the angular increase in the apparent elevation of a heavenly
body due to the refraction of light, and is always a negative correction.
Parallax (in altitude) is the angular decrease in the apparent elevation
of a heavenly body due to the observation being taken at the surface instead
of at the center of the earth, and is always a positive correction.
NOTATION
^= latitude ( when north, — when south);
Of = right ascension;
S= declination ( when north, — when south);
i = hour angle ( to west, — to east);
A = azimuth from north point ( when measured clockwise);
Z = azimuth from south point ( when measmed clockwise);
ft = altitude;
z=zenith distance;
r= refraction;
p= parallax.
168 GEODETIC SURVEYING
meridian local mean time is later than standard time, and vice
versa.
Example 1. New York, N.Y., uses 75thmeridian standard time. Given
the longitude of Columbia College as 73° 58' 24". 6 west of Greenwch, what
is the local mean time at 10'' 14™ 17^.2 p.m. standard time?
75° 00' 00".0 loll 14™ 173.2 p.m.
73 58 24 .6 4 06 .4
1 5) 1° 01' 35".4 Ans. =10'> 18™ 23^.6 p.m.
•i™ 06^.4
Example 2. Philadelphia, Pa., uses 75thmeridian standard time. Given
the longitude of Flower Observatory as 5^ 01™ 06^.6 west of Greenwich, what
is the standard time at 9'' 06™ 18^.1 a.m. local mean time.
15 )75° 00' 00".0 9'' 06™ 18M a.m.
Sh 00™ 00^0 1 06 .6
5 01 06 .6
Ans. =9'' 07™ 24''.7 A.M.
1™ 06^.6
97. Sidereal Time, In this kind of time a sidereal day of
twentyfour hours corresponds exactly to one revolution of the
earth on its axis, as marked by two successive upper transits
of any star over the same meridian. The sidereal day for any
meridian commences when that meridian crosses the vernal
equinox, and runs from zero to twentyfour hours. The sidereal
time at any moment is the hour angle of the vernal equinox at
that moment, counting westward from the meridian. As the
right ascensions of stars and meridians are counted eastward
from the vernal equinox, it, follows that the sidereal time
for any observer is the same as the right ascension of his
meridian at that moment. Hence when a star of known
right ascension crosses the meridian the sidereal time
becomes known at that moment. The right ascension
of the mean sun at Greenwich mean noon (called sidereal
time of Greenwich mean noon) is given in the Nautical
Almanac for every day of the year, and is readily found
for local mean noon at any pther meridian by adding the
product of 9.8565 seconds by the given longitude west of Green
v,dch expressed in hours.
ASTRONOMICAL DETERMINATIONS 169
98. To Change a Sidereal to a Mean Time Interval, and vice
versa. Owing to the relative directions in which the earth rotates
on its axis and revolves around the sun the number of sidereal
days in a tropical year (one complete revolution of the earth
around the sun) is exactly one more than the number of solar
days. According to Bessel the tropical year contains 365.24222
mean solar days, hence 365.24222 mean solar days = 366.24222
sidereal days, and therefore
1 mean solar day= 1.0027379 sidereal days;
1 sidereal day = 0.9972696 mean solar days;
whence if Is is any sidereal interval of time and Im the mean solar
interval of equal value, we have
/<, = /m + 0.0027379 /„, (log 0.0027379 = 7.4374176  10)
Im=Is  0.0027304 I, (log 0.0027304 = 7.4362263  10)
Where there is much of this work to be done the labor of computa
tion is lessened by usin^ the tables found in the Nautical Almanac
and books of logarithms.
99. To Change Local Mean Time or Standard Time to Sidereal.
For local mean time this is done by converting the mean time
interval between the given time and noon into the equivalent
sidereal interval (Art. 98), and combining the result with the
sidereal time of mean noon for the given place and date. Since
the right ascension of the mean sun increases 360° or twenty
four hours in one year, the increase per day will be 3™ 56^.555,
or 9^.8565 per hour. The sidereal time of mean noon for the
given place is therefore found by taking the sidereal time of Green
wich mean noon from the Nautical Almanac and adding thereto
the product of 9^8565 by the longitude in hours of the given
meridian, counted westward from the meridian of Greenwich.
If standard time is used it must first be changed to local mean
time (Art. 966) before applying the above rule.
Example. To find the sidereal time at Syracuse, N. Y., longitude
76° 08' 20" .40 west of Greenwich, when the standard (75th meridian) time
is 10" 42"! 00' A.M., January 17th, 1911.
170 GEODETIC SURVEYING
76° 08' 20". 40
75
IQH 42™ 00 '.00 standard time
 4 33 .36
15) 1° 08' 20". 40
4m 33=. 36
10 37 26 . 64 local mean time
12
log 4953.36 =3.6948999
log . 0027379 = 7 . 4374176
111 22" 338. 36 = 49533.36
+ 13 .56
log (13^56) =1.1323175
1 22 46 . 92 sidereal interva
15)76° 08' 20". 40
5h 04^338.36
log 9.8565 = 0.9937227
log 5.0759=0.7055131
= 5.0759 hrs.
log (503.03) = 1.6992358
Sidereal time of Greenwich mean noon 19'' 43™ 093.48
Reduction to Syracuse meridian + 50 .03
Sidereal time of Syracuse mean noon 19 43 59 . 51
Sid. int. from Syracuse mean noon — 1 22 46 .92
Sidereal time at given instant IS^ 21™ 12^. 59
100. To Change Sidereal to Local Mean Time or Standard
Time. This is the reverse of the process in Art. 99, and consists
in finding the difference between the given time and the sidereal
time of mean noon for the given place and date, changing this
interval to the corresponding mean time interval (Art. 98), and
combining the result with twelve o'clock (mean noon) by addi
tion or subtraction as the case requires. The result is local mean
time, and if standard time is wanted it is then obtained as
explained in Art. 966.
Example. To find the local mean time and standard (75th meridian)
time at Syracuse, N. Y., longitude 76° 08' 20".40 west of Greenwich, when
the sidereal time it 181 21™ 12s.59, January 17, 1911.
76° 08' 20".4075° = l° 08' 20". 40 = 4™ 333.36
log 9.8565 =0.9937227
15)76° 08' 20".40 log 5.0759 =0.7055131
5t 04m 33s. 36
= 5 . 0759 hrs. log (50^ . 03) = 1 . 6992358
ASTEONOMICAL DETERMINATIONS 171
Sidereal time of Greenwich mean noon 19'' 43™ 09^.48
Reduction to Syracuse meridian + 50 .03
Sidereal time of Syracuse mean noon 19 43 59 . 51
Sidereal time at given instant 18 21 12 . 59
Sidereal interval before Syracuse mean noon P 22™ 46^.92
Ih 22™ 46^.92 = 4966^92
log 4966. 92 =3.6960872
log . 0027304 = 7 . 4362263
log (133.56) =1.1323135
Reduction to 1 1" 22™ 46^92
mean time
interval
 13 .56
1 22 33 .36
12
Local mean time at given instant (mormng) 10'' 37™ 26^.64
Reduction to standard time +4 33 . 36
Standard time at given instant (morning) 10'' 42™ 00^.00
101. Time by Single Altitudes. The altitude of any heavenly
body as seen by an observer at a given point is constantly chang
ing, each different altitude corresponding to a particular instant
of time which can be computed if the latitude and longitude are
approximately known. In finding local mean time or sidereal
time it is sufficient to know "the latitude to the nearest minute
and the longitude within a few degrees. In changing from local
to standard time, however, an error of V will be caused by each
15" error of longitude. If the latitude is not known it may
generally be scaled sufficiently close from a good map, or it may
be determined as explained in Arts. 107 or 108. By comparing
the observed time for a certain measured altitude of sun or star
with the corresponding computed time the error of the observer's
timepiece is at once determined. The observation may be
made with a transit (or altazimuth instrument), or with a sextant
(and artificial horizon), the latter being the most accurate. In
either case several observations ought to be taken in imme
diate succession, as described below, and the average time and
average altitude used in the reductions. The probable error of
the result may be several seconds with a transit, and a second
or two with the sextant. The actual error is apt to be larger on
account of the uncertainties of refraction. The observation is
commonly made with the sextant and on the sun.
172 GEODETIC SURVEYING
101a. Making the Observation. The best time for making
an observation on the sun is between 8 and 9 o'clock in the
morning and between 3 and 4 o'clock in the afternoon, in order
to secure a rapidly changing altitude and at the same time avoid
irregular refraction as far as possible. The altitude of the
center of the sim is never directly measured, but the observations
are taken on either the upper or lower limb, or preferably an equal
number of times on each limb. Star observations may be made
at any hour of the night, selecting stars which are about three
hours from the meridian and near the prime vertical, and hence
changing rapidly in altitude at the time and place of observation.
If two stars are observed at about the same time having about
the same declination and about the same altitude, but lying on
opposite sides of the meridian, the mean of the two results (de
terminations of the clock error) will be largely free from the errors
due to the imcertainties of refraction.
In taking the observation an attendant notes the watch
time to the nearest second at the exact moment the pointing
is made. // the transit is used, an equal number of readings
should be taken with the telescope direct and reversed, the plate
bubble parallel to the telescope being brought exactly central
for each individual pointing in order to eliminate the instrumental
errors of adjustment. If a star or one limb of the sun is observed
there should be not less than 3 direct and 3 reversed readings.
If both limbs of the sun are observed there should be not less
than 2 direct and 2 reversed readings on each limb, or 3 direct
on one limb and 3 reversed on the other limb. If the sextant and
artificial horizon are used, and the pointings are made on a star
or on one limb of the sim, not less than 5 readings of the double
altitude should be taken; if both limbs of the sun are observed,
not less than 3 readings should be taken for each limb. These
double altitudes are always corrected for index error and some
times for eccentricity. It is considered better not to use the
cover on the artificial horizon, but if it has to be done it should
be reversed on half of the readings. If as much tin foil is added
to commercial mercury as it will unite with, an amalgam is formed
whose surface is not readily disturbed by the wind, thus rendering
the cover unnecessary. When the mercury is poured in its
dish it must be skimmed with a card to clean its reflecting surface.
In all of the above methods of observing, the work is supposed
ASTRONOMICAL DETERMINATIONS 173
to be C9,rried on with reasonable regularity and expedition when
once started. With any method it is desirable to take at least
two sets of readings and compute them independently as a check,
the extent of the disagreement showing the quality of the work
that has been done, while the mean value is probably nearer the
truth than the result of any single set.
101b. The Computation. The first step in the computation
of any set of observations is to find the average value of the meas
ured altitudes and the average value of the recorded times, these
average values constituting the observed altitude and time for
that set. This observed altitude is then reduced to the true
altitude for the center of the object observed. The reductions
which may be required are for refraction, parallax, and semi
diameter. The apparent altitude of all heavenly bodies is too
large on account of the refraction of Ught; Table VIII gives the
average angular value of refraction, which is a negative correc
tion for all measured altitudes. Parallax is an apparent dis
placement of a heavenly body due to the fact that the observer
is not at the center of the earth; star observations require no
correction for parallax; all solar observations require a positive
correction for parallax, the amount being equal to 8". 9 multiplied
by the cosine of the observed altitude. The correction for
semidiameter is only required in solar work, and not even then
for the average of an equal number of observations on both limbs;
when the average altitude refers to only one limb a selfevident
positive or negative correction is required for semidiameter,
the value of which is given in the Nautical Almanac for the me
ridian of Greenwich for every day of the year, and can readily
be interpolated for the given longitude. Letting h equal true
altitude for center, h' equal measured altitude, r equal refrac
tion, p equal parallax, and s equal semidiameter, we have
h (for a star) = h' — r;
h (sun, both limbs) = h' — r + p;
h (sun, one limb) = h' — r + p ± s.
In the polar triangle ZPS, Fig. 47, page 166, the three sides are
known. ZP, the colatitude, is found by subtracting the observer's
latitude from 90°. PS, the polar distance or codeclination, is
174 GEODETIC SUEVEYING
found by subtracting the declination of the observed body from 90°.
In the case of the sun the declination is constantly changing and
must be taken for the given date and hour (the time being always
approximately known). The sun's declination for Greenwich
mean noon is given in the Nautical Almanac for every day in the
year, and can be interpolated for the Greenwich time of the observa
tion; the Greenwich time of the observation differs from the
observer's time by the difference in longitude in hours, remember
ing that for points west of Greenwich the clock time is earlier, and
vice versa. ZS, the coaltitude, is found by subtracting the
reduced altitude h from 90°. Using the notation of Fig. 47,
we have from spherical trigonometry
cos 3 = sin ^ sin I? + cos ^ cos § cos t,
whence
cos z — sin + 3 = 63° 47' 35" .4
z+(^5) = 61 33 11 .2 z+{'p + 5)= 105 28 20 .0
z{(j>d) = 21 48 18 .0 z(^ + d) =~22 06 50 .8
tan i.= v ^ f °° f' y '•;; ' , ^!f '': ;r;! 21° 58' 37".7
^ V cos (52 44 10 .0) cos ( — 11 03 25 .4)
« = 43° 57' 15".4 = 2i» 55™ 49^.0.
Local apparent noon 12i> 00™ 00^0
Hour angle of sun — 2 55 49 .0
Apparent solar time 9^ 04™ 11^.0
Equation of time — 2 30 . 8
Local mean time of observation Q^ 01™ 40^.2
Watch time of observation 8 52 24 .0
Watch slow by mean time 9"^ 16^ . 2
Reduction to standard time — 9 33 .2
Watch fast by standard time 0™ 17= .
176 GEODETIC SURVEYING
In either case the error of the observer's timepiece (as deter
mined by any given set of observations) is obtained by comparing
the observer's average time for the given set with the computed
true time for the same set.
102. Time by Equal Altitudes. In this method the clock
time is noted at which the sun (or a star) has the same altitude
on each side of the meridian, from which the clock time of meridian
passage (upper or lower transit or culmination) is readily obtained.
By comparing the clock time with the true time of meridian
passage the error of the observer's clock is at once made known.
The advantages of this method over the method of single altitudes
are as follows: the results are in general more reliable; the com
putation is simpler, as it does not involve the solution of a spherical
triangle; no correction is required for refraction, parallax, semi
diameter, nor instrumental errors; the latitude need not be known
at all for star observations, and only very approximately for
solar work. The observations may be made with a transit or a
sextant (with artificial horizon), the latter being the most accurate.
In either case several observations ought to be taken in immediate
succession, as described below, and the average time used in the
reductions. The probable error of the result should not exceed
about two seconds with the transit nor about one second with
the sextant. The actual error may be greater on account of the
uncertainties of refraction. The method evidently assumes that
the refraction will be the same for each of the equal altitudes,
but on account of the lapse of time between the observations
this is not necessarily true. The observation is commonly made
with the sextant and on the sun.
102a. Making the Observation. As with the previous
method, the best time for making an observation on the sun is
between 8 and 9 o'clock in the morning and between 3 and 4
o'clock in the afternoon. The observations may be taken entirely
on one limb of the sun or an equal number of times on each limb.
The equal altitudes may be taken on the morning and afternoon
of the same day, or on the afternoon of one day and the morning
of the next day. For star observations a star should be selected
which will be about three hours from the meridian and near the
prime vertical at the times of observation. Since the equal
altitudes observed must be within the hours of darkness, a star
is required whose meridian passage occurs within about three
ASTRONOMICAL DETERMINATIONS 177
hours after dark and three hours before daylight. The sidereal
time of meridian passage is always known, since it is the same
as the star's right ascension, and the corresponding values of
mean time and standard time are readily found by Arts. 100 and
966. The equal altitudes may be taken during the same night, or
on the morning and evening of the same day.
In taking the observation the attendant notes the watch
time to the nearest second at the exact moment the pointing is
made. If the transit is used the telescope is not reversed, but the
plate bubble parallel to the telescope is brought exactly central
for each individual pointing; no corrections are made to the result
ing reading for any instrumental errors. If the sextant and
artificial horizon are used no corrections are applied to the result
ing double altitude as measured. There is no great objection
to using the cover of the artificial horizon in this method, and
when used it is not reversed (as in Art. 101a); it is necessary,
however, to use it in the same position at both periods of equal
altitudes.
If a star or one limb of the sun is observed there should be
not less than 5 readings taken at each period of equal altitudes.
If both limbs of the sun are observed there should be not less than
3 readings (at each period) for each limb. The angular readings
in this method are always equally spaced, the instrument being
set in turn for each equal change of altitude and the time noted
when the event occurs. In commencing operations the observer
measures the approximate altitude, sets his vernier to the next
convenient even reading, and watches for that altitude to be
reached; the next setting is then made and that altitude waited
for, and so on. At the second period the same settings must be
used, but in reverse order. The size of the angular interval
will depend on the abihty of the observer to make each setting
in time to catch the given occurrence, and can best be found by
trial; under average conditions a good observer would not find
it difficult to use 10' settings on the transit and 20' on the sextant.
It is desirable to take at least two independent sets of observa
tions, and compute them separately as a check and as an indica
tion of the reliability of the results; the adopted value would
then be taken as the mean of the several determinations.
102b. The Computation. In this method there is no object
in finding the average of the observed altitudes, the method
178 GEODETIC SURVEYING
being based on the equality of the corresponding altitudes in
stead of their value. For each set of observations, however,
it is necessary to find the average of the time readings for
each of the two periods of equal altitudes. From these values
the middle time (halfway point between the two average time
readings) is found for star observations, and the middle time and
elapsed time (interval between average time readings) for solar
observations. For star observations the middle time is the
observer's time of meridian passage. For solar observations
a correction must be applied to the middle time to obtain the
observer's time of meridian passage, on account of the changing
declination of the sun.
For solar observations on the same day, expressed in mean time
units, we have from astronomy
TT — M dd ■ t fisin. . The geodetic latitude can never be directly observed,
nor can the deviation of the plumb line be found by direct meas
ASTEONOMICAL DETEEMINATIONS
187
urement. If, however, the latitude of the point n be found
by computation (Chapter V) from the astronomical latitudes
measured at various other triangulation stations, and these
values be averaged in with its own astronomical latitude, the
result may be assumed to be free from the effects of plumb line
deviation and to represent the true geodetic latitude. In geodetic
work geodetic latitude is always understood unless otherwise
specified.
Geocentric latitude is the angle between the equator and the
radius vector from the center of the earth; in Fig. 49 the geo
centric latitude of the point n is the angle /?. The geocentric
188 GEODETIC SURVEYING
latitude can never be directly observed. It is computed from the
geodetic latitude by the formula
in which (Art. 69)
62
tan /? = "2 tan <{>,
log ^ = 9.9970504  10.
At the equator the geodetic and geocentric latitudes are each
equal to zero. At the poles they are each equal to 90°. At any
other point the geocentric latitude is less than the geodetic
latitude. By the calculus we have,
tan 96 (for ^  /? = max.) = ^, or (j> = 45° 05' 50".21;
tan /? (for 4>  ^ = max.) = , or /? = 44 54 09 .79;
or a maximum difference of 11' 40".42. The popular conception
of latitude is geocentric latitude, but published latitudes are
usually astronomical latitudes or geodetic latitudes.
107. Latitude from Observations on the Sun at Apparent
Noon. Latitude sufficiently close for many purposes may be
obtained by measuring the altitude of the sun at apparent noon,
or the moment when it crosses the meridian. The local mean
time of apparent noon is found by applying to 12 o'clock (the
apparent time) the equation of time as taken from the Nautical
Almanac for the given date, interpolating for the given meridian;
the corresponding standard time may then be found by Art. 96a.
If the correct time is not known the altitude is measured
when it attains its greatest value, which soon becomes evident
to the observer who is following it up. A good observer can obtain
an observation on each limb of the sun before there is any appre
ciable change of altitude, the mean of the readings being the
observed altitude for the center; if only one limb is observed
the reading must be reduced to the center by applying a correc
tion for semidiameter as found in the Nautical Almanac for the
given date, the result being the observed altitude. In either
case '(the observed altitude is too large on account of refraction,
and must be corrected by an amount which may be taken from
ASTRONOMICAL DETERMINATIONS 189
Table VIII for the given observed altitude. Theoretically all
solar altitudes are measured too small on account of parallax
(due to the observer not being at the center of the earth), the
necessary correction being equal to 8".9 multiplied by the cosine
of the observed altitude. The correction for parallax is a useless
refinement with the engineer's transit, but may be applied, if
desired, when a sextant or altazimuth instrument is used.
The observation. Single altitudes of the sun may be measured
with a transit or with an altazimuth instrument, but a pris
matic eyepiece will be required if the altitude exceeds about 60°.
The instrument must be very carefully leveled at the moment of
taking the observation, and if two readings can be secured the
second reading should be taken on the other limb of the sun with
the telescope reversed and the instrument carefully releveled,
so as to eUminate the instrumental errors. If only one reading
is seciu'ed it should be corrected for index error if one exists. If
the altitude is not greater than about 60° an artificial horizon
may be used and the double altitude measured with either of the
above instruments or a sextant. If a transit or altazimuth
instrument is used it is not reversed on any of the observa
tions, and it must not be releveled between the pointing to
the sun and the pointing to its reflected image. If a sextant is
used the correction for index error must be applied.
The computation. Having applied the appropriate correc
tions to the measured altitude, as described above, the true
altitude of the sun is obtained within the capacity of the instru
ment used. This value being subtracted from 90° gives the zenith
distance of the sun. The declination of the swa. is taken from the
Nautical Almanac for the given date and meridian, and this
value is the distance of the sun from the equator. Knowing thus
the distance from the equator to the sun, and from the sun to
the zenith, an addition or subtraction (as the case requires)
gives the zenith distance of the equator, and this value (Art. 106)
is the observer's latitude. If an ordinary transit is used the
latitude thus obtained should be correct to the nearest minute.
If a sextant or an altazimuth instrument is used the result is
generally much closer to the truth. Theoretically the result
should be as accurate as the instrument will read, but there is
always a doubt as to the precise value of the refraction, and the
latitude obtained is subject to the same uncertainty.
190 GEODETIC SURVEYING
108. Latitude by Culmination of Circumpolar Stars. Stars
having a polar distance (90° —declination) less than the observer's
latitude never set, but appear to revolve continuously around
the pole, and are hence called circumpolar stars. Such stars
cross the observer's meridian twice every day, once above the
pole (upper culmination) and once below the pole (lower culmina
tion). By referring to Fig. 47, page 166, it will be seen that the
latitude of any place is always the same as the altitude of the
elevated pole. By observing the altitude of a close circumpolar
star at either upper or lower culmination, and combining the
result (minus correction for refraction. Table VIII) with the star's
polar distance (added for lower culmination, subtracted for
upper culmination), the altitude of the elevated pole is obtained,
and hence the observer's latitude. The polar distance must be
based on the declination for the given date as found in the
Nautical Almanac. The latitude as thus determined is much
more reliable than that obtained by solar observations.
In the northern hemisphere the best star to observe is Polaris
(a Ursse Minoris), on account of its brightness (2nd magnitude)
and its small polar distance (about 1° 10' in 1911). About the
middle of the year both culminations of Polaris occur during
daylight hours, rendering it unsuitable for observation. The next
best star to observe is 51 Cephei, which also has a small polar dis
tance (about 2° 48' in 1911), but whose brightness (5th magnitude)
is not equal to that of Polaris. As these two stars differ about
five and onehalf hours in right ascension, at least one of them
must culminate during the hours of darliness. The sidereal time
of upper culmination for either star is the same as its right ascen
sion (the exact value for the given date being taken from the
Nautical Almanac), and this is converted into mean time by
Art. 100. By a study of Fig. 50, which shows the arrangement of
a number of stars in the vicinity of the north pole of the heavens,
it will not be difficult to identify Polaris and 51 Cephei. The
polar distances of these stars are so small that but little change
of altitude occurs when they are near the meridian, so that several
observations may be obtained and averaged. If the observations
are taken within five minutes each side of the meridian the error
in assuming the altitudes unchanging will not exceed 1" with
Polaris and 2".5 with 51 Cephei, and may be ignored when observ
ing with engineering instruments. Within fifteen minutes either
ASTRONOMICAL DETERMINATIONS 191
i
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•
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s> X^ "^o, •
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192 GEODETIC SURVEYING
way from meridian passage the change in altitude (within 1"
error) may be fomid, if desired, by multiplying the square of the
time (in minutes) from culmination by 0".044 for Polaris and
0".104 for 51 Cephei. If this correction is applied it is to be added
to observations near upper culmination and subtracted from
observations near lower culmination, to obtain the corresponding
culminating altitude.
In making the observation the altitude maybe directly measured
with a transit or an altazimuth instrument. In order to eliminate
instrumental errors at least two readings should be averaged
together, one taken with telescope direct and one with telescope
reversed. The instrument must be releveled after reversing,
as it is necessary to have the bubbles exactly central at the moment
each reading is taken. If by any accident only one reading is
secured it must be corrected for index error, if one exists. The
two readings should be obtained as near together and as near
culmination as the skill of the observer will permit; two readings
not over three minutes each way from the meridian are easily
obtained. A better result will be obtained if four readings are
averaged together, taking one direct reading, then two reversed
readings, and then one direct reading, both bubbles being kept
exactly central while taking each reading; this program is
easily accomplished within five minutes each side of the meridian.
If an artificial horizon is available it is better to measure the double
altitude between the star and its image in the mercury, using
either of the above instruments or a sextant. Angles measured
with a sextant are always correicted for index error and sometimes
for eccentricity. If a transit or altazimuth instrument is used the
double altitude is obtained by reading on the star and then on
its image, without reversing or releveling between the pointings.
Two such double altitudes are easily obtained within three minutes
each way from the meridian, using either of these instruments
or a sextant. Latitudes obtained by the methods of this article
should theoretically be correct within the reading capacity of
the instrmnent, but may be further in error on account of the
uncertainties of refraction.
109. Latitude by Prime Vertical Transits. Stars whose
declination is less than the observer's latitude apparently cross
the pHme vertical (true east and west vertical plane) twice dur
ing each revolution of the earth on its axis. If the time elapsing
ASTRONOMICAL DETERMINATIONS
193
between the east and west transit of any star is noted the observ
er's latitude may be found by com
putation. Referring to Fig. 51, F is
the elevated pole of the celestial
sphere; PZS', the observer's meridian;
Z, the observer's zenith; SZS", the
prime vertical; SS'S", the star's ap
parent path; PS, the star's polar
distance; and PZ, the observer's co
latitude. In the spherical triangle
PZS, rightangled at Z, the side PS
and the angle SPZ are known; the
side PS being the star's polar distance,
and the angle SPZ equal to half the
elapsed time changed to angular units by multiplying by 15.
Hence, solving for the latitude ^, we have
tan
= EZ = observer's latitude;
8 = ES = declination of S (from Nautical Almanac) ;
d' = ES' = declination of S' (from Nautical Almanac) ;
z = apparent zenith distance of S;
z'= apparent zenith distance of S';
r = refraction correction for z (from Table VIII) ;
r' = refraction correction for z' (from Table VIII) ;
then
z +r = ZS = true zenith distance of S;
z' + r' = ZS' = true zenith distance of iS';
whence
(f) = § + z + r
= d' {z'+r')
2?^ = (5 + d'} + {zz') + {rr'y
and we have for the latitude
^ = hlid + d') +{z z') + (r  r')\.
In this equation the quantities {d + 8') and (r — r') are known,
so that it is only necessary to obtaiu {z— z') by observation to
determine the latitude. The quantity {z — z') is the difference
between the zenith distances of the two stars S and S', and if
this quantity is not over about 15' it can be measured with great
accuracy by means of the zenith telescope (see Fig. 53). The
instrument illustrated has an aperture of about three inches,
a focal length of nearly four feet, and a magnifying power of 100.
The telescope being set at a proper vertical angle for a given pair
of stars is not changed thereafter, but each star is brought into
the field of view by revolving the instrument on its vertical
axis, and the difference of zenith distance is measured entirely
Fig. 53. — Zenith Telescope.
From a photograph loaned by the XJ. S. C. and G. S,
196 GEODETIC SURVEYING
with the micrometer eyepiece. Many pairs of stars are observed,
and many refinements in observation and computation are required
in the highest grade of work. For a complete discussion of the
method the reader is referred to Appendix No. 7, Report for
189798, U. S. Coast and Geodetic Survey. An altazimuth
instrument with a micrometer eyepiece will give very good
results by the above method, if used with proper precautions.
111. Latitude Determinations at Sea. Many methods have
been devised for determining latitude at sea. Greenwich time
may or may not be required, according to the method used,
but is generally available from the ship's chronometers. In any
case the observation consists in measuring with the sextant the
altitude of one or more of the heavenly bodies above the sea
horizon. All such altitudes are reduced to the true horizon by
applying a correction for dip, as explained in Art. 105, this cor
rection being in addition to any others which the observation
requires to determine the true altitude. The most common
observation for latitude is for the altitude of the sun at apparent
noon, as explained in Art. 107. The meridian altitude of the pole
star or other bright star is also often observed, the result in either
case being worked out as explained for circumpolar stars in
Art. 108. The error of a latitude determination at sea may range
upwards from a fraction of a mile, depending on the circumstances
surrounding the observation.
112. Periodic Changes in Latitude. It is now known that the
earth has a slight wabbling motion with respect to the axis about
which it rotates. In consequence of this motion the north and
south poles do not occupy a fixed position on the surface of the
earth, but each one apparently revolves about a fixed mean
point in a period of about 425 days. The distance between
the actual pole and the mean point is not constant, but varies
(during a series of revolutions) between about 0".16 (16.3 ft.),
and about 0".36 (36.6 ft.). As the equator necessarily shifts
its position ia accordance with the movement of the poles, it
follows that the latitude at every point on the smf ace of the earth
is subject to a continual oscillation about its mean value, the
successive oscillations being of different extent and ranging from
0".16 to 0".32 each way from the middle. In precise latitude
work, therefore, the date of the determination is an essential
part of the record.
ASTRONOMICAL DETERMINATIONS 197
Longitude
113. General Principles. The longitude of any point on the
surface of the earth is the angular distance of the meridian of that
point from a given reference meridian, being positive when reclconed
westward and negative when reckoned eastward. The meridian
of Greenwich has been universally adopted (since 1884) as the
standard reference meridian of the world, but other meridians
(Washington, Paris, etc.) are often used for special work. Since
time is measured by the uniform angular movement of the earth
on its axis (west to east), it follows that longitude may be
expressed equally well in either angular units or time units. As
360° of arc correspond to twentyfour hours of time (mean or
sidereal. Art. 95), the change from the angular to the time system
is evidently made by dividing by 15, and vice versa; thus the
longitude of Washington west from Greenwich may be written
as 77° 03' 56".7, or 5^ 08"" 15'.78, as preferred. At the same
absolute instant of time the true local time of any station differs
from the true local time of any other station by the angular
divergence (expressed in time units) of the meridians of these
two stations; the difference of longitude of any two stations,
therefore, is identical with the difference of local time. At the
same instant of time, the difference between the local mean time
and the sidereal time at any station is the same for all points in
the world, so that the difference of local time between any two
given stations is always numerically the same whether the com
parison is based on local mean time or sidereal time. From the
nature of the case, it is evident that standard time (Art. 96a)
bears no relation to the longitude of a station.
Longitude as described above is geodetic longitude. Longitude
obtained from observations on heavenly bodies, or astronomical
longitude, is identical with geodetic longitude except where local
deviation of the plumb line (Art. 75) exists. The geodetic long
itude of a point can never be directly observed, nor can the devia
tion of the plumb line be found by direct measurement. If,
however, the longitude of any point be found by computation
(Chapter V) from the astronomical longitudes measured at
various other triangulation stations, and these values be averaged
in with its own astronomical longitude, the result may be assumed
to be free from the effects of plumb line deviation and to represent
198 GEODETIC SUEVEYING
the true geodetic longitude. In geodetic work geodetic longitude
is always understood unless otherwise specified.
The longitude of any given point is ordinarily obtained by
finding how much it differs from that of some other point whose
longitude has already been well determined. The finding of
this difference of longitude is essentially the finding of the dif
ference of local time between the two points, the westerly
point having the earliest time, and vice versa. The local time
is found by the methods heretofore given, and the comparison
is made as about to be explained.
114. Difference of Longitude by Special Methods. These
methods are rarely used any more, but are of considerable scientific
interest, and hence are here briefly mentioned.
By special phenomena. Certain astronomical phenomena,
such as the eclipses of Jupiter's satellites, occur at the same instant
of time as seen at any point on the earth from which they may
be visible. These eclipses usually occur several times in the course
of a month, the Washington mean time of the event being given
in the Nautical Almanac. The observer notes the true local time
at which the eclipse occurs, the error and rate of his timepiece
having been previously determined. The difference between
the Washington mean time and the local mean time of the eclipse
is the observer's longitude from Washington. Eclipses of the
moon may also be used in the same manner. Longitude obtained
by these methods is apt to be several seconds of time in error,
or a minute or more in arc.
By flash signals. Two observers, having obtained their own
local time by proper observations, may each note the reading of
their own clock at the same instant of time, this instant being
determined by an agreed signal visible to both. Such a signal
may be the flash of a heliotrope by day, or any suitable fight
signal by night. The difference of local time is then the difference
of longitude. The error by this method may be kept below a
second of time by averaging the results of a number of signals.
This method usually requires one or more intermediate stations
to be established to overcome the lack of intervisibilityj and is
generally an expensive one.
115. Longitude by Lunar Observations. If an observer notes
his true local time (expressed as mean time) for any particular
position of the moon, and obtains from the Nautical Almanac
ASTEONOMIOAL DETERMINATIONS 199
the Greenwich mean time when the moon occupied suCh a posi
tion, the longitude from Greenwich is given by 'the corresponding
difference of time. Many methods have been devised on this
basis, requiring laborious computations in their application, and
in many of the methods not leading to very accurate results.
Lunar methods are therefore not generally used except on long
sea voyages or long exploration trips. A few of the methods are
given below, but only in the roughest outline.
By lunar distances. The angle between a star, the center
of a planet, or the near edge of the sun, and the illuminated edge
of the moon may be measured by a sextant, and reduced to
what it would have been if it had been observed at the center of
the earth and measured to the center of the moon. The Green
wich time of this position can be determined from the Nautical
Almanac and compared with the local time at which the observa
tion was made. The accuracy attainable is about five seconds
of^time.
By lunar culminations. The local time of meridian passage
of the moon's illuminated limb may be noted, expressed as sidereal
time and corrected for semidiameter, giving the moon's right
ascension at the given instant, and Greenwich mean time for
this value of the right ascension be compared with the observed
local time. The accuracy attainable is about five seconds of time.
By lunar occuUations. The occultation (covering) of a star
by the moon may be observed, noting the local time of immersion
(disappearance), or emersion (reappearance), or both, in which
case the apparent right ascension of the corresponding edge of
the moon at the given instant is the same as the right ascension
of the given star. When proper correction has been made for
refraction, parallax, semidiameter, etc., the true right ascension
becomes known for the given instant, and the corresponding
Greenwich time is compared as before with the local observed time.
This method, with the exception of telegraphic methods, is one of
the best that is known for longitude work. When a number
of such determinations are averaged together, an accuracy approx
imating a tenth of a second of time is attainable.
116. Difference of Longitude by the Transportation of Chro
nometers. When this method is used a number of chronometers
(from 5 to 50) are carried back and forth (from about 5 round trips
upwards) between the two points whose difference of longitude
200 GEODETIC SURVEYING
is desired. On reaching each station the traveling chronometers
are compared with the local chronometers. The errors of the
local chronometers are determined astronomically at or near
the time of comparison. The various values thus obtained for the
difference of time between the two stations are averaged together
and the result taken as the difference of longitude. Owing to
the fact that each round trip furnishes two determinations that
are oppositely affected by similar errors, and also to the refinements
of method and reduction that are used in practice, the errors
due to chronometer rates and irregularities are largely eliminated
from the average result. The accuracy attainable (in time imits)
may range between a few tenths of a second and less than a single
tenth of a second, depending on the distance between stations,
the number of trips made, and the number of chronometers
transported. Longitude determinations by this method are now
rarely made, except where telegraphic connection is not available.
In order to make an accurate comparison of two mean time
chronometers each one is independently compared with the same
sidereal chronometer, and two sidereal chronometers are sim
ilarly compared by mutual reference to a mean time chronometer.
Sidereal chronometers continually gain on mean time chronom
eters, the beats or ticks (half seconds) gradually receding from and
approaching a coincidence that occurs about every three minutes.
When the beats exactly coincide the chronometers differ precisely
by the value in half seconds indicated by the subtraction of their
face readings. As the ear can be trained to detect a lack of coin
cidence as small as the onehundredth part of a second, a com
parison can be made with this degree of precision.
117. Difference of Longitude by Telegraph. Where tele
graphic connection can be established between two stations it
furnishes the best means of exchanging time signals, both on
account of the great accuracy attainable and the comparative
inexpensiveness. Difference of longitude obtained in this manner
can be made more accurate than is possible by any other known
method. The lines of the telegraph companies ramify in all
directions, and the temporary use of a suitable wire can usually
be obtained at reasonable cost, so that it is only necessary to
erect short connecting lines between the observing stations and the
telegraph stations. The most important applications of the
method are as outlined below.
ASTRONOMICAL DETERMINATIONS / 201
By standard time signals. This method furnishes a quick
means for an approximate longitude determination. Standard
time can be obtained at any telegraph station with a probable
error of less than a second. The observer's true local mean time
is obtained by any of the simpler methods of observation. The
difference of these times is the difference of longitude between
the given standard time meridian and the meridian of the ob
server's station.
By star signals. The difference of longitude of any two
stations is identical with the sidereal time which elapses between
the transit of any given star over the meridian of the easterly
station, and the transit of the same star over the meridian of the
westerly station; so that it is only necessary to observe how long
it takes for any star to pass between the meridians of two stations
to know their difference of longitude. In making use of this
principle a chronograph (Art. 103c) is placed at each station,
and these chronographs are connected by a telegraph line. A
breakcircuit chronometer, which may be placed anywhere in
this line, records its beats on both chronographs. As the selected
star crosses the meridian of the easterly observer he records this
instant of time on both chronographs by tapping his break
circuit signal key. When the same star crosses the meridian of the
westerly observer he likewise records this new instant of time
on both chronographs. Each chronograph, therefore, contains
a record of the time between transits, but the records are not
identical, as it takes time for the signals to pass between the
stations; in other words, each signal is recorded a little later on
the distant chronograph than it is on the home chronograph.
The record of the easterly chronograph thus becomes too great,
and the record of the westerly one correspondingly too small;
but the mean of the two records eliminates this error and gives
(when corrected for chronometer rate) the true difference of
longitude between the stations. In actual work the transits of
many stars are observed at each station, so as to obtain an average
value for the difference of longitude. The accuracy attainable
is about 0.01 of a second of time. This method is one of the
best, and was formerly largely used by the Coast Survey. The
objection to the method is the difficulty of securing the monopoly
of the telegraph line during the long period while the observa
tions are in progress, so that it is no longer much in use.
202 GEODETIC SUEVEYING
By arbitrary signals. This is the standard method of the
Coast Survey at the present time, and requires the use of the
telegraph line for only a few minutes during an arbitrary period
(previously agreed upon) on each night that observations are in
progress. In this method a chronometer and chronograph are
installed at each station, and each chronometer records its beats
on the home chronograph only. Each observer makes his own
time observations, which are likewise recorded on his own chrono
graph alone. Observations at each station are taken both before
and after the exchange of signals in order to determine the cor
responding chronometer's rate as well as its error. As far as
possible the same stars are observed at each station, in order
to avoid introducing errors of right ascension. In the most
precise work the observers exchange places on different nights,
in order to eliminate the effects of personal equation, and numerous
other refinements are introduced. The chronograph sheet at
each station enables the true time at that station to be computed
for any instant within the range of the record, and the difference
of these true times at any one instant of time is the difference of
longitude between the stations. The whole object of the exchange
of signals, therefore, is to identify the same instant of time on
both chronograph sheets. At the agreed time for the exchange
of signals the two stations are thrown into circuit with the main
telegraph line, with connections so arranged that signals (momen
tary breaking of circuit) sent by either station are recorded on
both chronographs. No signal, however, is recorded at exactly
the same instant at both stations, on account of the time required
for its passage between them. The difference of longitude as
based on the signals from the western station is hence too large,
and that based on the eastern station's signals correspondingly
too small. The mean of the two values is taken as the true
difference of longitude, while the difference of the two values
represents double the time of signal transmission. In the Coast
Survey program two independent sets of ten pairs of stars
each are observed on five successive nights, the observers then
exchanging places and continuing the observations in the same
manner for five more nights. Signals are exchanged once each
night at about the middle time for the work of both stations,
the western station sending thirty signals at intervals of about
two seconds, followed by thirty similar signals from the eastern
ASTRONOMICAL DETERMINATIONS 203
station. These signals were formerly sent by the chronometers,
but are now sent by tapping a breakcircuit signal key. The
accuracy attainable, as in the case of star signals, is about 0.01
of a second of time.
118. Longitude Determinations at Sea. Every seagoing vessel
carries one or more chronometers, the error and rate of each being
determined before leaving port, so that the Greenwich time of
any instant is always very closely known. The local time for
the ship's position having been determined for any instant
(Art. 105), and the corresponding Greenwich time being obtained
from the chronometers, it is only necessary to take the difference
of these times to have the ship's longitude from Greenwich.
The result thus obtained is expressed in time units, but is readily
converted into angular units by multiplying by 15 (Art. 113).
In case of failure of the chronometers, longitude at sea can still
be determined in a number of ways not requiring a previous
knowledge of Greenwich time, such as the method of lunar dis
tances (Art. 115). Discussions and explanations of these methods
can be found in all works on Navigation and Nautical Astronomy.
A longitude determination at sea may be in error from a fraction
of a mile to a number of miles, depending on the surrounding
circumstances.
119. Periodic Changes in Longitude. As explained in Art. 112,
the poles of the earth are not fixed in position, but each one
apparently revolves about a mean point in a period of about 425
days, the radiusvector varying (during a series of revolutions)
between about 0".16 and 0".36. The result of this shifting of
the poles is to cause the longitude of any point to oscillate about
a mean value, the amplitude of the oscillation depending on the
location of the point. In precise longitude work, therefore, the
date of the determination is an essential part of the record.
Azimuth
120. General Principles. By the azimuth of a hne (or a
direction) from a given point is meant its angular divergence from
the meridian at that point, counting clockwise from the south
continuously up to 360°. From any intermediate point on a
straight line the azimuths towards the two ends always differ by
exactly 180°, so that in any case it is only necessary to determine
204 GEODETIC SURVEYING
the azimuth in one direction. In passing along a straight line
the azimuth varies continuously from point to point, unless the
line be the equator or a meridian. The cause of this change and
the methods for computing it are explained in detail in Arts. 68
to 73, inclusive. The following articles are concerned solely
with the determination of azimuth (and hence of the meridian)
at any one given point.
Geodetic azimuth is that in which the angular divergence from
the meridian is measured in a plane which is tangent to the
spheroid at the given point. Azimuth obtained from observations
on heavenly bodies, or astronomical az muth, is identical with
geodetic azimuth except where local deviation of the pliunb line
(Art. 75) exists. The geodetic azimuth of a line from a given
point can never be directly observed, nor can the deviation of
the plumb line be found by direct measurement. If, however,
the azimuth of a line from a given point be fovmd by computa
tion (Chapter V) from the azimuth determinations made at
various other triangulation stations, and these values be averaged
in with the observed value, the result may be assumed to be free
from the effects of plumb line deviation and to represent the true
geodetic azimuth. In geodetic work geodetic azimuth is always
understood unless otherwise specified.
121. The Azimuth Mark. This is the signal which gives the
direction of the line whose azimuth is being determined. An
azimuth mark should not be placed less than about a mile from
the observer, otherwise a change of focus will be required between
the heavenly body and the mark. Experience has shown that
refocussing during an observation is very undesirable. When
azimuth is obtained by solar observations any of the usual day
time signals (Art. 19) may be used, being located at a special
azimuth point or a regular triangulation station as circumstances
may require. When azimuth is obtained by stellar observations
a special azimuth point is generally located one or more miles
from the instrument. The azimuth mark should be moimted
on a post or otherwise raised about five feet above the ground,
and generally consists of a bull'seye lantern enclosed in a box
or placed behind a screen, a small circular hole being provided
for the light to pass through on its way to the observer. If the
diameter of the hole does not subtend over a second of arc (0.3
of an inch per mile) at the eye of the observer, the light will
ASTRONOMICAL DETERMINATIONS 205
closely resemble a star in both apparent size and brilliancy, which
is the object sought. The face of the box or screen is often painted
with stripes or other design so that it may also be observed in the
daytime.
122. Azimuth by Stm or Star Altitudes. The altitude of any
heavenly body as seen by an observer at a given point is con
stantly changing, each different altitude corresponding to a par
ticular azimuth which can be computed if the latitude and longi
tude are approximately known. For the degree of accuracy
sought by this method it is sufficient to know the latitude to the
nearest minute and the longitude within a few degrees. The
difference in azimuth of any two lines from the same point is
always exactly the same as their angular divergence. If, therefore,
the horizontal angle between the azimuth mark and the given
heavenly body is measured at the same moment that the altitude
is taken, the azimuth of the line to the azimuth mark is obtained
by simply combining the computed azimuth of the heavenly body
with this measured horizontal angle. The observation may be
made with a transit or an altazimuth instrument. The probable
error of a single determination should not exceed a minute of
arc with the ordinary engineer's transit, nor a half minute with the
larger instruments. The actual error may be larger than the
probable error on account of the uncertainties of refraction.
122a. Making the Observation. The best time for making
an observation on the sun is between about 8 and 10 o'clock in
the morning and 2 and 4 o'clock in the afternoon. The sun should
not be observed within less than two hours of the meridian
because its change in azimuth is then so much more rapid than
its change in altitude; nor when it is much more than four hours
from the meridian on account of the uncertain refraction at low
altitudes. In the latitude of New York it is not desirable to
observe the sun for azimuth in the winter time because its dis
tance from the prime vertical during suitable hours results in such
a rapid movement in azimuth as compared with its movement
in altitude. Star observations may be made at any hour of the
night, selecting stars which are about three hours from the meridian
and near the prime vertical, and hence changing but slowly in
azimuth as compared with the change in altitude. The observa
tions are made in sets of two, taking one reading with the tele
scope direct and the other with the telescope reversed, the mean
206 GEODETIC SURVEYING
horizontal and the mean vertical angle constituting the observed
values for that set. Several independent sets should be taken and
separately reduced, the mean of the resulting azimuths being the
most probable value. The instrument should be in perfect
adjustment and be leveled up with the long bubble or the striding
level, and should not be releveled except at the beginning of each
set. The center of the sun is not directly observed, but the read
ing is taken with the image of the sun tangent to the horizontal
and vertical hairs. A complete set is made up as follows: Sight
on the mark and read the horizontal circle; unclamp the upper
motion and bring the sun's image tangent to the horizontal and
vertical hairs in that quadrant where it appears by its own motion
to approach both hairs; note the time to the nearest minute and
read both circles; unclamp the upper motion, invert the telescope,
and bring the sun's image tangent in that quadrant where it appears
to recede from both hairs; note the time and read both circles;
unclamp the upper motion, sight on the mark and read the hori
zontal circle. A star set is taken in the same manner except that
in each pointing the image of the star is bisected by both hairs. If
the instrument does not have a full vertical circle the telescope
is not inverted between the observations, but an index correction
must be applied to the observed altitudes. The values used in the
computations of the next article are those which correspond to
the center of the observed object. If for any reason only one
observation is secured on the sun, thus leaving the set incomplete,
the observed altitude is reduced to the center by applying a
correction for semidiameter, and the observed horizontal angle
is reduced to the center by applying a correction found by divid
ing the semidiameter by the cosine of the altitude. The semi
diameter is taken from the Nautical Almanac for the given
time and date, and the correction is added or subtracted
in accordance with the particular limb of the sun which was
observed.
122b. The Computation. It is best to reduce each set inde
pendently and average the final results. The observed altitude
must first be reduced to the true altitude. The apparent altitude
of all heavenly bodies is too large on account of refraction, the
required correction being found in Table VIII. The apparent
altitude of the sun is also too small on account of parallax, the
amount being equal to 8". 9 multiplied by the cosine of the
ASTRONOMICAL DETERMINATIONS 207
observed altitude, but this correction is so small it would seldom
be applied in this method.
In the polar triangle ZPS, Fig. 47, page 166, the three sides are
known. ZP, the colatitude, is found by subtracting the observer's
latitude from 90°. PS, the polar distance or codeclination, is
found by subtracting the dechnation of the observed body from
90°. In the case of the sun the declination is constantly changing
and must be taken for the given date and hour (the time being
always approximately known). The sun's declination for Green
wich mean noon is given in the Nautical Almanac for every day
in the year, and can be interpolated for the Greenwich time of
the observation; the Greenwich time of the observation differs
from the observer's time by the difference in longitude in hours,
remembering that for points west of Greenwich the clock time
is earlier and vice versa. ZS, the coaltitude, is found by sub
tracting the true (reduced) altitude of the observed body from 90°.
Using the notation of Fig. 47, we have from spherical trigonometry,
sin d = cos z sin (f> + sin z cos (J) cos A,
whence
, sin d — cos z sin d)
cos A = ; J ,
sin z cos
d)]
tan iJL M^^^ ^^^ {4, + d)\ sin ^[z {<}> d)]'
The value of A thus found is the azimuth angle (from north
branch of meridian) of the given heavenly body at the moment
of observation. If the observed body was east of the meridian
its azimuth (from the south point) equals 180° + A ; if west of
the meridian, 180° — A. The azimuth of the azimuth mark is
then found by combining the azimuth of the observed body with
the corresponding angle between the azimuth mark and the
observed body, the combination being made by addition or
subtraction as the case requires.
123. Azimuth from Observations on Circumpolar Stars. The
simplest and most accurate method of determining azimuth is
by suitable observations on close circumpolar stars, furnishing
any desired degree of precision up to the highest attainable. In
208 GEODETIC SUEVEYING
northern latitudes the best available stars are a Ursse Minoris
(2nd magnitude), d Ursee Minoris (4th magnitude), 51 Cephei
(5th magnitude), and A Ursae Minoris (6th magnitude). Of these
four a UrsiE Minoris, commonly known as Po aris, is usually
chosen by engineers on account of its brightness, the other three
being barely visible to the naked eye. The four stars named may
be identified by reference to Fig. 50, page 191.
Owing to the rotation of the earth on its axis the azimuth
of any star, as seen from a given point, is constantly changing,
but the value of the azimuth may be computed for any given
instant of time when the position of the observer is known. The
most favorable time for the observation of a close circumpolar
star is at or near elongation (greatest apparent distance east or
west of the meridian), as its motion in azimuth is then reduced
to a minimum; but entirely satisfactory results may be obtained
from observations taken at any time within about two hours
either way from elongation; the only point involved is that time
must be known with increasing accuracy the greater the interval
from elongation, in order to secure the same degree of precision
in the azimuth determination. In any case the actual observation
consists in measuring the horizontal angle between an azimuth
mark and the given star, and noting the time at which the star
pointing is made. The azimuth of the mark is then obtained
by combining the measured angle (by addition or subtraction
as the case requires) with the computed azimuth of the star.
The details of the observation will depend on the instrument
available and the degree of precision desired, in the result. The
instruments used may be the ordinary engineer's transit, the
larger transits equipped with striding levels, the repeating instru
ment, or the direction instrument. Close instrumental adjust
ments are necessary for good work. The methods ordinarily
used are the direction method, the repeating method, and the
micrometric method. Certain formulas enter more or less into
all the methods.
123a. Fundamental Formulas. The following symbols are
involved in the formulas as here given:
A = azimuth of star (at any time) from north point,
+ when east, — when west ;
Ae = azimuth of star at elongation;
ASTEONOMICAL DETERMINATIONS 209
Ao = azimuth of star at mean hour angle of n pointings;
n = number of pointings to star;
t = hour angle of star (at any time), + when star is
west, — when east, or may be counted westward up
to 24 hours or 360°;
te = hour angle of star at elongation;
M = interval of any one hour angle from the mean of
n given hour angles;
C = curvature correction in seconds of arc;
D = correction for diurnal aberration La seconds of arc;
De = ditto for a close circumpolar star at elongation;
4> = latitude, + when north, — when south;
S = declination of star, + when north, — when south;
Am = azimuth of mark from north point, + to east,
— to west;
Z = azimuth of mark from south point;
h = mean altitude of star;
d = value of one division of bubble tube in seconds;
w, w', etc. = readings of west end of bubble tube when sighting
on star;
W = mean value of w, w' , etc. ;
e, e', etc. = readings of east end of bubble tube when sighting on
star;
E = mean value of e, e', etc. ;
■ 6 = mean inclination of telescope axis in seconds when
sighting on star;
X = angle correction in seconds due to inclination of
telescope axis;
oi = star's right ascension;
8 = sidereal time at any instant;
Se = sidereal time of star's elongation.
a. Hour angle at any instant. The hour angle of a star
(in time units) at any instant of sidereal time is given by the
formula
t=S a.
The corresponding value of t in angular units is obtained
(Art. 95) by multiplying by 15. The particular unit in which t is
to be expressed is always apparent from the formula in which it
occurs. If local mean time or standard time is used it must be
210 GEODETIC SURVEYING
reduced to sidereal time (Art. 99) before being used in the formula
for t. .
b. Hour angle at elongation. In the polar triangle ZPp,
Fig. 47, page 166, p may be taken to represent Polaris or any
other star at elongation, or greatest apparent distance from the
meridian for the observer whose zenith is at Z. In this triangle
the side PZ is the observer's colatitude, the side Pp is the star's
codeclination, and the angle ZpP equals 90° on account of the
tangency at the point p. Solving for the angle ZPp, or the star's
hour angle at elongation, we have
tan d)
cos te = 7 f.
tan
c. Time of elongation. Having found te from the formula
in (&), the sidereal time of elongation is given by the formulas
Se = a + te (western elongation),
Se = ct — te (eastern elongation).
The sidereal time thus obtained is changed to local mean time or
standard time by Art. 100 when so desired.
d. Azimuth at elongation. If the above triangle (6) be solved
for the angle PZp, or the star's azimuth at elongation, we have
. sin polar distance cos 8
sm Ae = 1 ,., J = X
cos latitude cos p
e. Reduction to elongation. If the angle between the azimuth
mark and a close circumpolar star is measured within about
thirty minutes either way from elongation, 'the measured angle
may be reduced very nearly to what it would have been if measured
at elongation by applying the following correction:
, 2 sin2 i{te  t)
A.  A = ta,ii Ae ^ '
sin 1"
The quantity (ie ~ is equivalent to the sidereal time interval
from elongation, and may be substituted directly without com
puting the hour angle represented by t. If the mean or standard
ASTEONOMICAL DETEEMINATIONS 211
time interval is thus used the value which the formula gives for
(Ae— A) must be increased by tIit P^'^ of itself.
/. Azimuth at any hour angle. If the star is observed at any
other hour angle than that which corresponds to elongation, a polar
triangle will be formed similar to ZPp, Fig. 47, page 166, but
with all the angles oblique. In this case the azimuth A at the
given hour angle t is given by the formula,
. sin t
tan A =
sin ^ cos t — cos ^ tan d
_ cot d sec (j) sin t
1 — cot d tan ^cos t
= — cot d sec ^ sin tlr. 1,
in which
a = cot d tan ^ cos t.
g. The curvature correction. If a series of observations are
taken on a star the hour angle and corresponding azimuth must
necessarily be different for each pointing. The mean value of
such azimuths is frequently desired, and may of course be found
by computing each azimuth separately and averaging the results.
The same value, however, may be obtained much more simply
by computing the azimuth corresponding to the mean of the
several hour angles, and then applying the socalled curvature
correction to reduce this result to the mean azimuth desired.
The reason that such a correction is required is because the motion
of a star in azimuth is not uniform, but varies from zero at elonga
tion to a maximum a1 culmination. In the case of a close circum
polar star, and a series of observations not extending over about
a half hour, the curvature correction is given by the formula
„ ^ . 1 „ 2 sin2 iJt
C = tan 4oS . J, ,
n sm 1
in which Jt is expressed in angular value, or
C = tan 4o^^ sin l"5:(ii)2
2 n '
212 GEODETIC SURVEYING
in which Jt is expressed in sidereal seconds of time. If Jt is
expressed in meantime seconds the value of C thus obtained
must be increased by rhr part of itself.
log— ^ sm 1"
= 6.7367275  10.
The sign of the curvature correction C is known from the fact that
the true mean azimuth always lies nearer the meridian than the
azimuth that corresponds to the mean hour angle. From the
nature of the case it is evident that the several values of M in
time units may be obtained directly from the observed times
(without changing them to hour angles) by taking the differences
between each observed time and the mean of all the observed times.
h. Correction for inclination of telescope axis. If the axis
of the telescope is not horizontal the line of sight will not describe
a vertical plane when the telescope is revolved on this axis, and
hence the measured angle between the star and the mark will be
in error a corresponding amount. The inclination of the axis
is found from the readings of the striding level. If the level is
reversed but once the usual formula is
b =^[{w+w') (e + e')];
but if the level is reversed more than once it is more convenient
to write
b=^iWE).
So far as the present purpose is concerned these formulas are
equally applicable whether the level is actually reversed on the
pivots, or reversed in direction because the instrument is turned
through 180°. In one case the value obtained is the actual
average inclination of the axis, while in the other case it is the
net inclination. By the east or west end of the bubble tube is
meant literally the end which happens to be east or west when the
reading is taken. The correction required on account of the
inclination b, due to the altitude of the star, is
X = b tan h.
ASTRONOMICAL DETERMINATIONS 213
The value of x thus obtained is to be subtracted algebraically
from the computed azimuth of the mark. Ordinarily a similar
correction for inclination due to altitude of mark is not required,
as the mark is generally nearly in the horizon of the instrument.
If, however, the angular elevation (+ altitude) or depression
( — altitude) of the mark is reasonably large, the striding level
should be read when pointing to the mark and a similar correction
computed. In this case the correction is to be added algebraically
to the computed azimuth of the mark.
i. Correction for diurnal aberration. Owing to the rotation
of the earth on its axis and the aberration of light thereby caused,
the apparent position of any star is always more or less east of
its true position, the amount of the displacement depending on
the position of the observer and the position of the star. A
corresponding correction is required for all azimuths based on
the measurement of a horizontal angle between a mark and a
star, and is given by the formula
cos h
which for a close circumpolar star at elongation reduces to
Z)e=0".32 cos A.
In obtaining azimuth from a north circumpolar star it is evident
that the azimuth of the mark (counting clockwise from either
the north or south point) must be increased by the amount of
the above correction.
j. Reduction of azimuth to south point. In making azimuth
determinations by observations on north circumpolar stars it is
customary to refer all results to the north point until the azimuth
of the mark is thus expressed. The azimuth of the mark from the
south point is then given by the formula
Z = 180° + Am,
in which proper regard must be had to the negative sign of A „ if
it is taken counterclockwise.
123b. Approximate Determinations. It is frequently desirable
to make approximate determinations of azimuth, either because
the work in hand does not call for any greater accuracy, or as a
preliminary to the more accurate location of the meridian. Such
214 GEODETIC SURVEYING
determinations may be made by measuring sun or star altitudes,
as explained in Art. 122, but observations on Polaris (or other
circumpolar stars) give more reliable results without any increase
in either field or office labor. The ordinary engineer's transit
may be used for such work, and with proper care will give correct
results within the smallest reading of the instrument. Since
the observation is best made at or near elongation the time of
elongation (c, Art. 123a) is computed beforehand, so that proper
preparation may be made. Assuming the instrument to be in
good adjustment and carefully leveled, the observation consists
in reading on the mark with telescope direct, reading on the star
with telescope direct, reading on the star with telescope reversed,
and ending with a reading on the mark with telescope reversed.
The lower motion must be left clamped and all pointings made with
the upper motion alone. The instrument must not be releveled
during the set. Both plate verniers should be read at each pointing.
The four pointings should be made in close succession, but with
out undue haste or lack of care. If the observation is being made
at elongation the first pointing to the mark is made a few minutes
before the computed time of elongation, and the two star point
ings as near as may be to the time of elongation. If time is not
accurately known the star is followed with the telescope until
elongation is evidently reached, when the necessary observations
are quickly taken. For five minutes each side of elongation the
motion of the star in azimuth is scarcely perceptible in an engineer's
transit. If the observations are not taken at elongation time must
be accurately known and read to the nearest second at each star
pointing. The observations having been completed the mean
angle , between the mark and the star is obtained from the four
readings taken, and it only remains to compute the mean azimuth
of the star to know the azimuth of the mark. If the star point
ings were made within about ten minutes either way from elonga
tion the azimuth of the star may be taken as equal to its azimuth
at elongation (rf. Art. 123a). If the star pointings were made
within about a half hour either way from elongation the angle
between the mark and the star may be reduced to what it would
have been at elongation by use of the formula for reduction to
elongation (e. Art. 123a), the quantity (te—t) being taken as
the angular value of the time interval between the time of elonga
tion and the average time of the star pointings. If the observa
ASTEONOMICAL DETERMINATIONS 215
tions are taken over about a half hour from elongation it is better to
compute the true aziinuth of the star for the average time of the
star pointings (/, Art. 123a).
123c. The Direction Method. In this method the angle
between the mark and the star is measured with a direction
instrument (Arts. 4247), the process being substantially the same
as there described for measuring angles between triangulation
stations. Owing to the fact that the star is in motion during the
observations, however, the angle being measured is constantly
changing, and the reductions must be correspondingly modified.
Owing to the altitude of the star serious errors are introduced
by any lack of horizontality in the telescope axis, and a cor
responding correction muSt be made in accordance with the read
ings of the striding level. If the mark is more than a few degrees
out of the horizon a similar correction will be required for the same
reason. The observations may be made at any hour angle, good
work requiring time to be known to the nearest second. A good
program for one set is to read twice on the mark with telescope
direct; then read twice on the star with telescope direct, noting
the exact time of each pointing and the reading of each end of
the striding level at each pointing; then read twice on the star
with telescope reversed, noting time and bubble readings as
before; then read twice on the mark with telescope reversed.
The striding level is left with the same ends on the same
pivots throughout the observations. Thb mean azimuth of the
star for the four pointings is then found by computing the
azimuth corresponding to the average time of these pointings
(/, Art. 123a), and then applying the curvature correction
(g, Art. 123a). The apparent azimuth of the mark is then found by
combining the computed star azimuth with the mean measured
angle. The true azimuth of the mark (as given by this set) is
then found by applying to the apparent azimuth the level cor
rection and the aberration correction {h and i, Art. 123a), and
reducing the result to the south point (j, Art. 123a). By taking
a nimiber of sets each night for several nights, and averaging
the different results, a very close determination of azimuth
may be secured. With skilled observers the probable error of a
single set should not exceed about a half a second of arc, and this
may be reduced to a tenth of a second by averaging about twenty
five sets.
216
GEODETIC SUEVEYING
EXAMPLE.— AZIMUTH BY DIRECTION METHOD *— RECORD
Station: Mount Nebo, Utah.
Instrument: 20m. Theodolite No. 5.
Star: Polaris, near lower culmination.
Date: July 21, 1887.
Observer: W. E.
Position X.
Object.
Chron.
Time.
Pos.
of
Tel.
Mic.
Circle Heading.
Forw.
d.
Back,
d.
Mean
d.
Corr.
for
Run.
Cor'd
Mean
Levels and
Remarks.
k, m.
Az. mark
Az. mark
Star
Star
Star
Star
Mean of
4 times
Az. mark
Az. mark
15 06 47.0
IS 10 23.3
IS IS 57.8
15 19 41.8
15 13 12 4
140
136
53
53
09
11
15
14.8
14.6
32.3
25.6
14.2
13.4
29.7
20.6
19.1
14.7
14.4
32.1
14.2
13.5
30.0
20.4
19.2
45.3
44.3
60.7
43.0
43.8
59.2
50.1
48.7
07.0
07.2
22.6
06.5
06.3
21.0
12.3
11.3
41.3
32.0
44.0
40.5
30.3
43.7
39.1
38.2
09.5
57.5
10.5
08.5
57.3
10.0
05.8
05.3
27.0
17.8
29.0
26.0
16.5
27.5
24.6
23,3
28.3
18.7
29.7
26.5
16.7
28.7
24.0
19.8
19.8
49.4
11.8
38.6
24.0
24.8
0.2
0.5
19.6
19.6
0.2
10.5
0.2
0.2
12.1
38.4
06.1
23.8
"\V. B.
43.5 27.0
53.7 17.5
97.2 44.5
452.7
39.5 32.3
27.4 44.6
66.9 76.9
10.0
Mean circle
reading:
On star:
136''12'26".3!
On mark;
140°63'21".90
* Abridged from example ia Appendix No. 7, Report for 189798, U. S. Coast and
Geodetic Survey.
ASTRONOMICAL DETEEMINATIONS
217
AZIMUTH BY DIRECTION METHOD— COMPUTATION
Mount Nebo, Utah, July, 1887.
= 39° 48' 33". 44
* Explanation.
Date and position
July 21, X
July 21, XI
Mean chronometer time
IShlSm 12^.44
0'»55m 10=. 06
Chronometer correction
35 .40
34 .62
Sidereal time
15 12 37 .04
54 35 .44
a of polaris
1 17 58 .16
1 17 58 .48
t of polaris (time)
13 54 38 .88 '
0 23 23 .04
t of polaris (arc)
208 °39' 43". 20
550' 45". 60
d of polaris
88 42 06 . 13
88 42 06 .20
log cot S
8.35532
8.35532
log tan
9.92087
9.92087
log cos t
logo
9.94323 n
9.99773
8.21942 n
8.27392
log cot d
8.355325
8.355319
log sec
0.114537
0.114537
log sin t
9.680917 n
9.007983 n
log 1/1 a
log (—tan A)
9.992861
0.008237
8.143640 n
7.486076 n
A
+0° 47' 51". 02
+0° 10' 31". 68
6"25s,4 81".
7'°08^8 100". 3
, 2 sin^ iJt
2 49 .2 15 .6
3 23 .1 22 .5
J
o
CO
^QJ
"5
1*
a
f
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to^
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<^ +
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CSrH
•*.+
■^ Oi Oi »OrH t^
00 .i
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t> rH OS O
irid
03
>H
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1— 1
f*
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6
"Is
Iz;
ll
rH (N CO "*
lO CO
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IN
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lO
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1
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<
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rt
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rt Q
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aj"
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(N lO
t^
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00
ril 115
Tf
^
00
!i
CO O "5 1H
TH lO
iH
'J*
1i
lO rH
ri
cos t
sin cos t
cos ^ tan d
cos ^ tan 3 — sin <(> cos i . . . .
log sin i
log (cos (fi tan 5 — sin ^ cos t)
log ( — tan A)
A
Jt and
2 sin' yt
sinl" ■
June 6
14h54m 17s. 7
31 .1
14 53 46 .6
1 21 20 .3
13 32 26 .3
203° 06' 34". 5
9.73876
9.96367 n
9.70243 n
 0.5040
+ 38.7399
+ 39.2439
9.593830 n
1 . 593772
8.000058 n
+0° 34' 22". 7
7m47s,7
5 09 .7
1 26 .7
1 52 .3
4 54 .3
7 37 .3
119". 3
52 .3
4 .1
6 .9
47 .2
114 .0
1 2 sin2 i^t
n sm 1"
log (curvature correction)
Curvature correction
Mean azimuth of star. . . .
Angle starmark
Level correction
Corrected angle
Azimuth of mark E. of N
343 .S
57 .3
1.7582
9.7583
+ 0.6
+ 0°34' 22". 1
72 57 50 .2
 1 .6
48 .6
73 32 10 .7
June 6
15hll" 48^2
31 .1
15 11 17 .1
1 21 20 .3
13 49 56 .8
207° 29' 12".
9.73876
9.94798 n
9.68674 n
 0.4861
+ 38.7399
+ 39.2260
9.664211 n
1 . 593574
8.070637 n
0°40' 26". 9
7m04s,2
4 30 2
1 54 .2
2 26 .8
4 25 .8
6 35 .8
98". 1
39 .8
7 .1
11 .8
38 .5
85 .4
280 .7
46 S
1.6702
9.7408
+ 0.6
+ 0° 40' 26". 3
72 51 46 .7
 1 .8
44 .9
73 32 11 .2
ASTRONOMICAL DETERMINATIONS 221
123e. The Micrometric Method. In this method the angle
between the mark and the star is measured with an eyepiece
micrometer, no use whatever being made of the horizontallimb
graduations. Any form of transit or theodohte rnay be used
that contains an eyepiece micrometer arranged to measure
angles in the plane defined by the optical axis and the horizontal
axis of the telescope. An eyepiece micrometer is essentially
the same as the micrometer found on the microscopes of direc
tion instruments and described in Art. 45. When the observing
telescope is fitted with an eyepiece micrometer the moving hairs
lie in the focal plane of the objective and pass across the images
of the objects viewed. When the angle between two' objects is
small (about two minutes or less) it may be assumed with great
exactness to be proportional to the distance between the corre
sponding images in the telescope, and this distance is measured
by the micrometer screw with great precision. In applying this
method to the determination of azimuth the mark is placed nearly
in the vertical plane through the star, and the small horizontal
angle between the mark and the star is determined from measure
ments made entirely with the micrometer, leaving all the hori
zontal motions of the instrument clamped in a fixed position.
The azimuth of the mark is then obtained by combining this
angle with the computed azimuth of the star.
In the eyepiece micrometer the value of the angle measured
is not given directly by the readings taken, as these indicate
only the number of revolutions made by the screw. The reading
is commonly taken to the nearest thousandth of a revolution,
the whole number of revolutions being read from the comb scale,
the t(3nths and hundredths from the graduations on the head,
and the thousandths by estimation. In order to convert the read
ing into angular value it is necessary to know the angular value
of one turn of the micrometer screw. The value of one turn of
the screw is foimd by measuring therewith an angle whose value
is already known. The value of such an angle may be found by
measuring it directly with the horizontal circle, or by computing
it from linear measurements. The value of one turn of the screw
piay also be obtained by observations on a close circumpolar star
near culmination, since the angle between any two positions of
the star is readily computed from the times of observation, and the
necessary reductions are then easily made.
222 GEODETIC SURVEYING
As already stated, the eyepiece micrometer measures angles
in the_ plane defined by the optical axis and the horizontal axis
of the telescope, and the corresponding horizontal angle must
hence be obtained by a suitable reduction for the given altitude.
To measure the horizontal angle between two objects at different
elevations, therefore, it is necessary to find the micrometer value
for the distance of each, object from the line of coUimation, reduce
each value to the horizontal for the corresponding altitude, and
combine the results for the complete horizontal angle. The reduc
tion in each case is effected by multiplying the micrometer value
by the secant of the altitude. In the case of azimuth determina
tions the reduction must necessarily be made for the star, but
need not be made for the mark unless it is several degrees out of
the horizon.
The micrometric method may be used at any hour angle,
but unless the star is near elongation it will pass out of the safe
range of the micrometer after but two or three sets of observa
tions have been secured. If the mark is placed about one or
two minutes nearer the meridian than the star at elongation,
the observations may be carried on within an hour or more each
way from elongation, and a small error in time will have little
or no effect on the result. In Coast Survey Appendix No. 7,
Report for 189798, the following procedure is recommended:
" The micrometer line is placed nearly in the line of coUimation of
the telescope, a pointing made upon the mark by turning the
horizontal circle, and the instrument is then clamped in azimuth.
The program is then to take five pointings upon the mark;
direct the telescope to the star; place the striding level in posi
tion; take three pointings upon the star with chronometer times;
read and reverse the striding level; take two more pointings upon
the star, noting the times; read the striding level. This com
pletes a halfset. The horizontal axis of the telescope is then
reversed in the wyes; the telescope pointed approximately to
the star; the striding level placed in position; three pointings
taken upon the star with observed chronometer times; the strid
ing level is read and reversed; two more pointings are taken
upon the star, with observed times; the striding level is read; and
finally five pointings upon the mark are taken." In reducing
such a set of observations the micrometer reading for the line of
coUimation is taken as the mean of all the readings on the mark,
ASTRONOMICAL DETERMINATIONS
223
05
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224 GEODETIC SURVEYING
AZIMUTH BY MICROMETRIC METHOD— COMPUTATION
CoUimatlon reads J(18 . 3^34 + 18 . 2808) = 18' . 2971
Markeastofcollimation, 18.313418.2971 =0.0163= 02". 02
Circle E., star E. of collimation
(18. 404218. 2971)(1. 1690)= .1252
Circle W., star E. of collimation
(18. 297118. 0912)(1. 1695)= .2408
Mean, star E. of collimation = 0.1835= 22 .70
Mark west of star = 20 . 68
Level correction (1 . 55) (0 . 92) (0 . 606) =  . 86
Mark west of star, corrected = 19 . 82
Mean chronometer time of observation = 211" 10™ 36^ .6
Chronometer correction =—2 11 28.2
Sidereal time = 18 59 08 . 4
a = 1 20 07 .4
Hourangle, t, m time 17^' 39™ 01= .0
Hourangle, i, in arc 264° 45' 15".
log cot d
log tan 96
log cos t
=
8.34362
9.78436
8.96108 n
log a
log cot d
log sec ^
log sin t
log 1/1 a
=
7.08906 n
8.343618
0.068431
9.998177 n
9.999467
log ( — tan A)
A
log 12.67
=
8.409693 n
+ 1° 28' 16"
1 . 10278
.91
log curvature correction
Curvature correction
Diur. aber. corr.
=
9.51247
0
+0
.33
32
Mean azimuth of star
Mark west of star
=
+ 1° 28' 16"
19
.90
.82
Azimuth of mark, E. of N. = + 1° 27' 57" .08
ASTEONOMICAj. •!• , . ' n N vTIONS 225
and all micrometer readings are rett ^ue. Since the
star is changing rapidly in altitude th. ter readings
are reduced to the horizontal for the meu .' ai fach half
set, the altitude of the star being occasioii r. '.:■ inter
polated for any desired time. The mean a^ i ' ' r tar
for each set is found by computing the azimuth < K^,
to the average time of the pointings (/, Art. 123a), a.
the curvature correction (gr. Art. 123a). The apparent
of the mark is then found by combining the computed star aZi. ii
with the measured angle (reduced to the horizontal). Theti <
azimuth of the mark (as given by this set) is finally found by apply
ing to the apparent azimuth the level correction and the aberra
tion correction (/i and i, Art. 123a), and reducing the result to the
south point (j, Art. 123a).
The time occupied in taking a set of observations in the man
ner above specified should not average over fifteen minutes,
so that a number of sets may be taken in a single night. By
averaging the results of a number of nights' work a very close
determination of azimuth may be secured. The method is more
accurate than the direction method or the repeating method.
With skilled observers the probable error of the mean of 25 or 30
sets should be less than a tenth of a second.
124. Azimuth Determinations at Sea. It is sometimes neces
sary to make an azimuth determination at sea in order to test
the correctness of the ship's compasses. The method commonly
employed is to measure the altitude of the sun or one of the brighter
stars, and at the same instant take its bearing as shown by the
compass to be tested. The azimuth of the given heavenly body
is then computed from its observed altitude and the result reduced
to a bearing. The difference between the observed bearing and
the computed bearing is the error of the compass. The method
and reductions for the azimuth observation are the same as
explained in detail in Arts. 122, 122a, and 1226, except that the
observation consists in measuring the altitude above the sea
horizon by means of a sextant, and that a correction for dip
(Art. 105) must be made. The latitude and longitude of the ship's
position are always sufficiently well known for use in the reduc
tions. The computed bearing should not be in error over a few
minutes, which is very much closer than it is possible to take the
compass bearing.
226 ■ IBk KEYING
SBflb^
125.
Pe
c CU
the pole!
^
k; ear^n »
entlv
:A ak/',''
.tzimuth. As explained in Art. 112,
■ ,<; fixed in position, but each one appar
xiean point in a period of about 425 days,
ib svecior ying (during a series of revolutions) between
/uui 0".'* '**' j"36. The result of this shifting of the poles
iB to a»'i#  azimuth of a line from a given point to oscillate
aboiJ*" p in value, the amplitude of the oscillation depending
OB i' ycation of the point. In precise azimuth work, therefore,
■:t, ate of the determination is an essential part of the record.
CHAPTER VIII
GEODETIC MAP DRAWING
126. General Considerations. The object of a geodetic u. fs or
chart is to represent on a flat surface, with as much accuracy ■ f
position as possible, the natural and the artificial features of a given
portion of the earth's surface. It is presumed that the engineer
is familiar with the lettering of maps and the usual methods of
representing the natural or topographical features, and such mat
ters are not here considered. The artificial features of a map
are the meridians and parallels, the triangulation system or other
plotted lines of location, and any lines which may be drawn to
determine latitude, longitude, azimuth, angles, distances, or areas.
In an absolute y perfect map the meridians and other straight
lines (in the surveying sense), would appear as straight lines; the
meridians would show a proper convergence in passing towards
the poles; the parallels of latitude would be parallel to each other
and properly spaced, and would cross all meridians at right angles;
all points would be properly plotted in latitude and longitude; and
azimuths, angles, distances and areas would everywhere scale
correctly. On account of the spheroidal shape of the earth, it
is evident that such a map is an impossibility, except for very
limited areas. Some form of distortion must necessarily exist
in any representation of a double curved surface on a flat sheet.
By selecting a type of projection depending on the use to be made
of the map, however, the distortion may be minimized in those
features where accuracy is most desired, and entirely satisfactory
maps produced. The principal types of map projection, as
explained in the following articles, are the cylindrical, the trape
zoidal, and the conical, these terms referring to the considerations
governing the plotting of the meridians and parallels.
In the work of plane surveying the areas involved are usually
of such small extent that no appreciable error is introduced in
plotting by plane angles and straight line distances, drawing all
227
/
/
228 GEODE'^i' Si i NG
/
meridians or other nr «tsul ■ ^^ lines perfectly straight and
parallel, and all pp " oil. ^/6ast and west lines also straight
and parallel anr' ,^s with the meridians. On account
of the larp y/ed in geodetic work it is generally
necessary ; ;,. , it VcA /fidians and parallels first (in accordance
with f 'f'' >i' ji4 of projection and the scale of the map),
and !. , >, 1 1 .iyfiindamental point of the survey by means of
i' . !.' ^ yix)ngitude without regard to angles or distances.
■■ihf. > "la ^cfetails may then be plotted as in plane surveying.
li% .
atthelat. ^ j 180(1  e^ sin^ 96)*'
in which formulas the letters have the significaiice and values of
Arts. 67 and 69. The values of one degree of latitude and longitude
are given for a number of latitudes in Table IX, and may be
interpolated for intermediate latitudes.
Since the radius of curvature of the meridian section increases
from the equator to the poles it follows that the above formula
for the length of a degree of latitude can only be correct in the
immediate vicinity of the given latitude. The true length L
of a meridian arc extending from the equator to any latitude 4>
is given by the formula
L = a(l  e)2(Af^ ~ N sm2(j) + P sin A.4,  etc.),
in which
ill = 1 + e2 + l^e* +. . .,
N =fe2 + Me* +. . .,
p = a^e^ + . . ■ •
For the length I of a meridian arc from the latitude
' ~ sin 40)].
GEODETIC MAP DRAWING
229
Substituting the values of a and e from Art. 67, and reducing
the formula to its simplest form, we have
I = A{(j)'  )  B sin (96'  ' + 0)
+ C sin 2 (^'  (j)) cos 2(^' + 96),
in which ' in the first term of the second member are to
be expressed in degrees and decimals, and in which
A =
B =
metric, 111133.30
feet, 364609.84
metric, 32434.25
feet, 106411.37
„ _ ( metric,
\ feet.
34.41
112.89
log A =
log 5 =
logC =
metric, 5.0458443
feet, 5.5618285
4.5110039
5.0269881
metric
feet,
metric, 1.5366847
feet, 2.0526689
127. Cylindrical Projections. The distinguishing feature of
all cylindrical projections consists in the projection of the given
area on the surface of a right cylinder (of special radius) whose
axis is the same as the polar axis of the earth. The flat map
desired is then produced by the development of this cylinder.
In all forms of this projection the meridians are projected by the
meridional planes into the corresponding right line elements of
the cylinder, so that after development the meridians appear as
equidistant parallel straight lines. The parallels of latitude
are projected into the circular elements of the cylinder in a nim:iber
of different ways, but in any case, after development, appear as
parallel straight lines crossing the meridians everywhere at right
angles. The three most common types of this projection are
explained in the following articles.
127a. Simple Cylindrical Projection. In this type of pro
jection, as illustrated in Fig. 54, page 230, the cylinder is so taken
as to intersect the spheroid at the middle latitude of the area to be
mapped, the parallels of latitude being projected into the cylinder
by lines taken normal to the surface of the spheroid. It is evident
from the figure that the parallels will not be represented by equi
distant lines, but will separate more and more in advancing towards
the poles. This distortion in latitude is offset to a certain extent
by a similar error in longitude, caused by the lack of convergence
in the plotted meridians, so that the various topographical features
remain approximately true to shape. On account of the varying
230
GEODETIC SURVEYING
distortion in both latitude and longitude no single scale can be
correctly applied to all parts of such a map. For the true lengths
of one degree of latitude or longitude see Table IX or Art. 126.
The projected distance x between the meridians, per degree of
longitude, due to the middle latitude ', is given by the formula
_ 7ra r cos ' 1
~ 180 [ (1  e2 sin2 <6')* J '
and the projected distance y, from the equator to any parallel
4>, by the formula
y=atan^[ ^^_J^.^;^,^, ] 
ae^ sin )^'
X
X
X
X
1
X
I
I
Fig. 54. — Simple Cylindrical Projection.
in which formulas the letters have the significance and values
of Arts. 67 and 69.
When the cylinder is taken tangent to the equator (making
(1 — e^ sin^ ^)i'
In making a map by this method the meridians and parallels
are spaced in accordance with the above formulas, and the fimda
mental points of the survey are then plotted by latitudes and
longitudes. For small areas (10 square miles) within about 45°
of the equator there is not much distortion in such a map. The
amount of the distortion in any case is readily obtained by com
GEODETIC MAP DRAWING
231
paring the results given by the true formulas and the formulas
used for the projection.
127b. Rectangular Cylindrical Projection. In this type of
projection, as illustrated in Fig. 55, the cylinder is so taken as to
intersect the spheroid at the middle latitude of the area to be
mapped, and the meridians are correctly developed on the ele
ments of the cylinder, so that in the finished map the parallels
are spaced true to scale. The error due to the lack of convergence
of the meridians still remains, so that the same scale can not be
applied to all parts of the map. The distortion in longitude is more
apparent than in the preceding projection, because no distor
tion exists in latitude. As in the previous case the meridians
are spaced true to scale along the central parallel.
•^daieLat.0'
Equator
X
X
X
X
X
X
Fig. 55. — Rectangular Cylindrical Projection.
In makiug a map by this method the central meridian and
parallel are first drawn and graduated to scale, using Table IX
or the formulas of Art. 126. The remaining parallels and meridians
are then drawn, and the survey plotted by latitudes and long
itudes. For small areas (10 square miles) within about 45° of
the equator there is not much distortion in such a map, straight
lines on the ground being straight on the map, and angles and
distances scaling correctly. The plotting for such an area may
therefore be done by latitudes and longitudes, or by angles and
distances, as in plane surveying.
127c. Mercator's Cylindrical Projection. This type of pro
jection, which is largely used for nautical maps, is illustrated
in Fig. 56, page 232. As in the simple cylindrical projection,
the space between the parallels constantly increases in advancing
from the equator towards the poles, but the spacing is governed
by an entirely different law. In Mercator's cylindrical projec
tion the cylinder is taken as tangent at the equator, so that the
232
GEODETIC SUEVEYINa
spacing of the meridians along the equator is true to scale in the
finished map. As the plotted meridians fail to converge, the
distance between them is too great at all other points, the extent
of the distortion becoming more and more pronounced as the
latitude increases. To offset this condition the distance between
the parallels is also distorted more and more as the latitude
increases, making the law of distortion exactly the same in both
cases. In that part of the map where the distance between the
meridians scales twice its true value, for instance, the distance
between the parallels should also scale twice its true value.
Since this distortion factor changes with the slightest change of
^
/
/
/
/
/
— ~~^ / ''

k
/
Any Lat,0\^''
\J
1
V
/
\
1
1
'
Eguator
X
X
X
X
X
1
1
\
Fig. 56. — ^Mercator's Cylindrical Projection.
latitude, however, it is evident that a satisfactory map will require
the meridian to be built up of a great many very small pieces,
each multiplied in length by its own appropriate factor. A per
fect map on this basis requires an infinitesimal subdivision of the
meridian, and a summation of these elements by the methods of
the integral calculus. Using the notation and the formulsis of
Arts. 67 and 69, and remembering that the distortion of any
parallel is inversely proportional to its radius, we have for the
distortion factor s at any latitude 0,
a
r
(1
N cos i
sina^)^
cos ^
Multiplying the meridian element, Rd4>, by the distortion factor
s, we have for dy, the projected meridian element.
dy = s{Rd)
o(l  e2)#
cos ^(1 — e^ sin2(/))4'
GEODETIC MAP DRAWING 233
whence, by integration,
.I.I5..8.5a[l„.(i±)e.og([±^)],
in which y is the projected distance from the equator to any
parallel of latitude 4^, and in which the formula is adapted to the
use of common logarithms. The value of x per degree of longitude,
for the spacing of the meridians, is given by the formula
_ 7ca
In making a map by this method the meridians and parallels
are spaced in accordance with the above formulas, and the fun
damental points of the map are then plotted by latitudes and
longitudes. It is evident that such a map will be true to scale
only in the vicinity of the equator, and that different scales must
be used for every part of the map. If it is desired, however, to
have the map true to any given scale along the central parallel ^',
it is only necessary to divide the above values of x and y by the
distortion factor s' corresponding to the latitude ^'.
A rhumb line or loxodrome between any two points on a spheroid
is a spiral line which crosses all the intermediate meridians at the
same angle. Except for points very far apart such a line is not
very much longer than the corresponding great circle distance.
Great circle sailing is sometimes practised by navigators, but
ordinarily vessels follow a rhumb line, keeping the same course
for considerable distances. A rhumb line of any length or angle
will always appear in Mercator's projection as an absolutely
straight line, crossing the plotted meridians at exactly the same
angle as that at which the rhumb line crosses the real meridians.
When a ship sails from a known point in a given direction, there
fore, its path is plotted on a Mercator chart by simply drawing a
straight line through the given point and in the given direction.
The distance traveled by the ship is plotted in accordance with the
scale suitable to the given part of the map. Similarly the proper
course to sail between any two points can be scaled directly from
the map with a protractor. It is for these reasons that this type
of projection is so valuable for nautical purposes.
234
GEODETIC SURVEYING
128. Trapezoidal Projection. In this type of projection,
as illustrated in Fig. 57, the meridians and parallels form a series
of trapezoids. All the meridians and parallels are drawn as
straight lines. The central meridian is first drawn and properly
graduated in degrees or minutes. The parallels of latitude are
then drawn through these points of division as parallel lines at
right angles to this meridian. Two parallels, at about onefourth
and threefourths the height of the map, are then properly gradu
ated, and the corresponding points of division coimected by a series
of converging straight lines to represent the meridians. For
the correct distances required in making the graduations see
Graduated
Correctly
Graduated
Correctly
Graduated
Correctly
Fig. 57. — Trapezoidal Projection.
Table IX or Art. 126. From the nature of the construction it is
plain that the central meridian is the only one which the parallels
cross at right angles. The fundamental points of such a map
are plotted by latitudes and longitudes. For small areas
(25 square miles) the distortion in distance is very slight in
this type of map.
129. Conical Projections. The distinguishing feature of the
conical projections consists in the projection of the given area
on the surface of one or more right cones (of special dimensions)
whose axes are the same as the polar axis of the earth. The
flat map desired is then pr duced by the development of the
cone or c nes thus used. In some forms of this projection the
meridians are projected into the right line elements of the cones,
while in other forms a different plan is adopted ; so that in some '
forms the meridians become straight lines after development,
GEODETIC MAP DEAWING
235
while in other forms they appear as curved lines. The parallels
of latitude are always projected into the circular elements of the
cone or cones, and after development always appear as circular
arcs. The four most common types of this projection are explained
in the following articles.
129a. Simple Conic Projection. In this type of projection,
as illustrated in Fig. 58, the projection is made on a single cone
taken tangent to the spheroid at the middle latitude of the area
to be mapped. The meridians are projected into the right line
elements of the cone by the meridional planes, and appear as
straight lines after development. The meridians are correctly
developed on the elements of the cone, so that the parallels are
all spaced true to scale on the finished map, The parallels are
A
A
^^\b
/Middle Lat.(>'
\C
\d
I Equator
^\
Fig. 58. — Simple Conic Projection.
drawn as concentric circles from the center A, the distance AC
being the tangent distance for the middle latitude. The central
parallel is graduated true to scale, and the meridians are drawn as
straight lines from the center A through the points of division.
For the tangent distance AC we have, from Art. 69,
a cot
AC = T = N cot (f>
(1  e2 sin2 ^)i
The correct values for graduating the meridian and central
parallel may be taken from Table IX or computed by the formulas
of Art. 126.
When it is impracticable to draw the arc EH from the center
A it may be located by rectangular coordinates from the point
C, as indicated by the dotted lines. To find the coordinates of
236 GEODETIC SUKVEYINa
any point H (see Fig. 59) let d equal the angular difference of
longitude subtended by the arc CH (radius = r), and 8' equal the
developed angle subtended by the same arc CH (radius =iV cot. ).
Then, since equal lengths of arc in different circles subtend angles
inversely as the radii, we have
d' _ r _ N cos (}) _ . ,
I ~ N cot
~ ^^'^ ^'
giving
8'= d sm(f>;
whence
X = AH sin S'= N cot
),
and
y =^AH vers d'= 2N cot ^ sw?(d^^^\.
These values of x and y are readily computed by means of the
data given in Table IX. In this projection the coordinates of
the different arcs vary directly as their radii,
so that the coordinates of the remaining parallels
may be found by a simple proportion. As a
check on the work the meridians should be
straight and uniformly spaced.
In making a map by this method the merid
ians and parallels are spaced in accordance with
the above rules, and the fundamental points of
the survey are then plotted by latitudes and
longitudes. In this projection the meridians and
Fig. 59. parallels intersect at the proper angle of 90°, and
the parallels are properly spaced; but the spacing
of the meridians is exaggerated everywhere except along the
central parallel, and all areas are oo large. Such a map is satis
factory up to areas measuring several hundred miles each way.
129b. Mercator's Conic Projection. In this type of pro
jection, as illustrated in Fig. 60, the projection is made on a single
cone, taken so as to intersect the spheroid midway between the
middle parallel and the extreme parallels of the area to be mapped.
The remaining parallels may be considered as projected into the
cone so that the spacing along the line BF is exactly proportional
to the true spacing along the meridian GHK; or mathematically
BC _CD_ ^ _ chord CE
GC ~ CH ~ arc CE '
GEODETIC MAP DRAWING
237
After development the entire figure is then proportionately
enlarged until the spacing of the parallels is again true to scale;
following which the developed angle and its subdivisions are
correspondingly reduced in size, in order to make the projected
parallels C'C" and E'E" true to the same scale. The distances
B'C = arc GC, CD' = arc CH, etc., are found from Art. 126 or
Table IX. The radius A'C is then computed from the formula
A'C _ cos il e^ sin2 ^")i
A'C + arc CE cos (j)" (1  e^ sin2 ^)i"
The remaining radii are found from A'C by a proper combina
tion of the known distances along the line A'F'. The parallel
A A'
Fig. 60. — ^Mercator's Conic Projection.
E'E" is then graduated both ways from the central meridian by
means of the values found from Art. 126 or Table IX, and the
mer dians are drawn as straight lines from the point A'.
The parallels may be plotted by rectangular coordinates
when it is impracticable to use the center A', but the values given
in Table IX are not correct for this type of projection. The
individual angles at the apex A' are readily obtained from the
radius A'E' and the subdivisions along the arc E'E", and the
coordinates are then found for this arc and proportioned for the
other arcs directly as their radii.
In making a map by this method the meridians and parallels
are drawn in accordance with the above rules, and the fundamental
238
GEODETIC SURVEYING
points of the survey are then plotted by latitudes and longitudes.
In this projection the meridians are straight lines, the meridians
and parallels cross at the proper angle of 90°, and the parallels
of latitude are properly spaced. The meridians are properly
spaced on the parallels C'C" and E'E", but are a little too widely
spaced outside of these parallels, and a Uttle too closely spaced
within these parallels. Areas outside of these same parallels are
too large, while areas within them are too small; but the total
area is nearly correct. Mercator's conic projection is suitable
for very large areas, having been used for whole continents. It
has also been largely used for the maps in atlases and geographies.
129c. Bonne's Conic Projection. In this type of projection,
as illustrated in Fig. 61, the projection is made on a single cone
A
/^^
■ \\n
/Middle Lat.?i'
\n
\e
\\f
1 Equator
1 '
/
/
/
/
/
/
/
qAV
E ^
F _
Fig. 61 — Boime's Conic Projection.
taken tangent to the spheroid at the middle latitude of the area
to be mapped. The central meridian is projected into the straight
line AF, with the parallels spaced true to scale and drawn as
concentric circles, in accordance with the rules and formulas for
simple conic projection (Art. 129a). Each parallel is then gradu
ated true to scale (see Art. 126 or Table IX), and the meridians
are drawn as curved lines through corresponding divisions of the
parallels.
In making a map by this method the fundamental points of
the survey must be plotted by latitudes and longitudes. In this
projection the meridians and parallels fail to cross at right angles
GEODETIC MAP DRAWING
239
but the same scale holds good for all the meridians and all the
parallels. Bonne's conic projection is suitable for very large
areas, having been used for whole continents. It has also been
largely used for the maps in atlases and geographies.
129d. Polyconic Projection. In this type of projection, as
illustrated in Fig. 62, a separate tangent cone is taken for each
parallel of latitude, and made tangent to the spheroid at that
parallel. Each parallel on the map results from the development
of its own special cone, appearing as the arc of a circle, with a
radius equal to the corresponding tangent distance. The parallel
Fig. 62. — Polyconic Projection.
through the point G, for instance, is drawn as a circular arc with
a radius equal to the tangent distance BG, and so on. The
central meridian is drawn as a straight line, on which all the
parallels are spaced true to scale, so that the division EF equals
the arc EF, the division FG equals the arc FG, and so on. The arcs
representing the various parallels are then drawn through these
division points with the appropriate radii, and with the centers
located on the central meridian. Each parallel as thus represented
is then graduated true to scale, and the meridians are drawn as
curved lines connecting the corresponding divisions.
In making a map by this method the meridians and parallels
are plotted in accordance with the data given in Table IX, or
240 GEODETIC SURVEYING
from corresponding values computed by the rules and formulas
of Arts. 126 and 129a, remembering that each parallel is here
equivalent to the central parallel of the simple conic projection.
The plotting is customarily done by rectangular coordinates,
the meridians and parallels being taken so close together that the
intersection points may be connected by straight lines. The
fundamental points of the survey are then plotted by latitudes
and longitudes.
This type of projection is suitable for very large areas. The
meridians are spaced true to scale throughout the map and cross
the parallels nearly at right angles. The parallels are spaced
true to scale only along the central meridian, and diverge more
and more from each other as the distance from the central merid
ian increases. The whole of North America, however, may be
represented without material distortion. The U. S. Coast and
Geodetic Survey and the U. S. Geological Survey have adopted
the polyconic system of projection to the exclusion of all others.
For further information on this subject see " Tables for the
Projection of Maps, Based upon the Polyconic Projection of
Clarke's Spheroid of 1866, and computed from the Equator to
the Poles; Special Publication No. 5, U. S. Coast and Geodetic
Survey, U. S. Government Printing Office, 1900."
The above type of polyconic projection is sometimes called
the simple polyconic, to distinguish it from the rectangula/ poly
conic, in which the scales along the parallels are so taken as to
make all the meridians and parallels cross at right angles. When
not otherwise specified the simple polyconic is in general under
stood to be the one intended.
PART II
ADJUSTMENT OF OBSERVATIONS BY THE
METHOD OF LEAST SQUARES
CHAPTER IX
DEFINITIONS AND PRINCIPLES
130. General Considerations. In various departments of
science, such as Astronomy, Geodesy, Chemistry, Physics, etc.,
numerous values have to be determined either directly or indirectly
by some process of measurement. When any fixed magnitude,
however, is measured a number of times imder the same apparent
conditions, and with equal care, the results are always foimd to
disagree raore or less amongst themselves. With skillful observers,
and refined methods and instruments, the absolute values of
the discrepancies are decreased, but the relative disagreement
often becomes more pronoxmced. The conclusion is obviously
reached that all measurements are affected by certain small and
unknown errors that can neither be foreseen nor avoided. The
object of the method of Least Squares is to find the most probable
'values of imknown quantities from the results of observation,
and to gage the precision of the observed and reduced values.
131. Classification of Quantities. The quantities observed
are either independent or conditioned.
An independent quantity is one whose value is independent of
the values of any of the associated quantities, or which may be
so considered during a particular discussion. Thus in the case
of level work the elevation of any individual bench mark is an
independent quantity, since it bears no necessary relation to the
elevation of any other bench mark. While in the case of a triangle
241
242 GEODETIC SUEVEYING
we may consider any two of the angles as independent quantities
in any discussion in' which the remaining angle is made to depend
on these two.
A conditioned quantity (or dependent quantity) is one whose
value bears some necessary relation to one or more associated
quantities. In any' case of conditioned quantities we may regard
these quantities as being mutually dependent on each other, or
any number of them as being dependent on the remaining ones.
Thus if the angles of a triangle are denoted by x, y, and z, we
may write the conditional equation
x + y + z = 180°,
and regard each angle as a conditioned quantity; or we may write,
for instance,
z = 180° X y,
and regard z as conditioned and x and y as independent.
132. Classification of Values. In considering the value of
any quantity it is necessary to distinguish between the true value,
the observed value, and the most probable value.
The tru£ value of a quantity is, as its name implies, that value
which is absolutely free of all error. Since (Art. 130) all measure
ments are subject to certain unknown errors, it follows that the
true value of a quantity may never be known with absolute pre
cision. In any case such a value would seldom be any exact
number of units, but could only be expressed as an unending
decimal.
The observed value of a quantity is technically understood to
mean the value which results from an observation when correc
tions have been applied for all known errors. Thus in measuring
a horizontal angle with a sextant the vernier reading must be
correcited for the index error to obtain the observed value of the
angle; in measuring a base line with a steel tape the corrections
for horizontal and vertical alignment, pull, sag, temperature, and
absolute length, are understood to have been applied; and so on.
The most probable value of a quantity is that value which is
most likely to be the true value in view of all the measurements
on which it is based. The most probable value in any case is
not supposed to be the same as the true value, but only that value
which. is more likely to be the true value than any other single
value that might be proposed.
DEFINITIONS AND PRINCIPLES 243
133. Observed Values and Weights. The observations which
are made on unknown quantities may be direct or indirect, and
in either case of equal or of unequal weight.
A direct observation is one that is made directly on the quan
tity whose value is desired. Thus a single measurement of an
angle is a direct observation.
An indirect observation is one that is made on some function
of one or more unknown quantities. Thus the measurement
of an angle by repetition represents an indirect observation,
since some multiple of the angle is measured instead of the single
value. So also in ordinary leveling the observations are indirect,
since they represent the difference of elevation from point to
point instead of the elevations of the different points.
By the weight of an observation is meant its relative worth.
When observations are made on any magnitude with all the con
ditions remaining the same, so that all the results obtained may
be regarded as equally reliable, the observations are said to be of
equal weight or precision, or of unit weight. When the condi
tions vary, so that the results obtained are not regarded as equally
reliable, the observations are said to be of unequal weight or pre
cision. It has been agreed by mathematicians that the most
probable value of a quantity that can be deduced from two obser
vations of unit weight shall be assigned a weight of two, from three
such observations a weight of three, and so on. Hence when an
observation is made under such favorable circmnstances that the
result obtained is thought to be as reliable as the most probable
value due to two observations which would be considered of unit
weight, we may arbitrarily assign a weight of two to such an
observation; and so on. As the weights apphed in any set of
observations are purely relative, their meaning will not be changed
by multiplying or dividing them all by the same munber. The
elementary conception of weight is therefore extended to include
decimals and fractions as well as integers, since any set of wei2,hts
could be reduced to integers by the use of a suitable factor.
134. Most Probable Values and Weights. In any set of
observations the most probable value of the unknown quantity
will evidently be some intermediate or mean value. There are
many types of mean value, but manifestly they are all subject
to the fundamental condition that in the case of equal values the
mean value must be that common value. Three of the common
244 GEODETIC SURVEYING
typms of mean value are the arithmetic mean, the geometric mean,
and the quadratic mean. If there are n quantities whose respective
values are Mi, M2, etc., we have,
= the arithmetic mean;
n
'^MiM2 . . . Mn = the geometric mean;
= the quadratic mean;
S
(1)
all of which satisfy the fundamental condition of a mean value.
In the case of direct observations of equal weight it has been
universally agreed that the arithmetic mean is the most probable
value. In accordance with this principle, and the definition of
weight as given in Art. 133, it is evident that the weight of the
arithmetic mean is equal to the number of observations. Sim
ilarly, an observation to which a weight of two has been assigned
may be regarded as the arithmetic mean of two component obser
vations of unit weight, and so on, provided no special assimiption
is made regarding the relative values of these components.
For direct observations of unequal weight, therefore.
Let z = the most probable value of a given magnitude;
Ml, M2, etc. = the values of the several measurements;
Pi, P2, etc. = the respective weights of these measurements;
api, ap2, etc. = the corresponding integral weights due to the
use of the factor a;
f^i', mi", etc. = the api imit weight components of Mi when con
sidered as an arithmetic mean
m2', m2", etc. = similarly for M2, and so on;
then we may write as equivalent expressions
,, mi' + mi" . . . Smi
Ml = =
api api
whence
^T wi2' I m2" . . . 2m2 ,
jy12 = = , etc;
ap2 ap2
Smi = apiMi,
Sm2 = ap2M2, etc.;
DEFINITIONS AND PRINCIPLES 245
and, since the various values of m are of unit weight,
_ Smi + Sm2 . . .
api + ap2 . . . '
or
^ Sap Sap ~ Sp ' • ■ • ■ ^'
from which we have the general principle :
In the case of direct observations of unequal weight the most
probable value is found by multiplying each observation by its weight
and dividing the sum of these products by the sum of the weights.
The result thus obtained is called the weighted arithmetic mean.
In the above discussion the value of z is found by taking the
arithmetic mean of Sap quantities whose sum is Sm, so that the
integral weight of z is Sap. Dividing by a in order to express this
result in accordance with the original scale of weights, we have
Weight of 3 = Sp; ....... (3).
or, expressed in words, the weight of the weighted arithmetic
mean is equal to the sum of the individual weights.
135. True and Residual Errors. It is necessary to distin
guish between true errors and residual errors.
A true error, as its name implies, is the amount by which any
proposed value of a quantity differs from its true value. True
errors are generally considered as positive when the proposed
value is in excess and vice versa. Since (Art. 132) the true value
of a quantity can never be known, it follows that the true error
is likewise beyond determination.
A residual error is the difference between any observed value
of a quantity and its most probable value, in the same set of
observations. The subtraction is taken algebraically in which
ever way is most convenient in the given discussion. In the case
of indirect observations the most probable value of the observed
quantity is found by substituting the most probable values of the
individual unknowns in the given observation equation (Art. 158).
Residual errors are frequently called simply residuals.
In the case of the arithmetic mean the sum of the residual errors
is zero. This is proved as follows:
246
GEODETIC SURVEYING
Let
Ml, M^,
Vl, V2,
then
n = the number of observations;
Mn = the observed values;
2 = the arithmetic mean;
. . Vn= the residual errors;
vi = z — Ml
V2 = z — M2
but
or
from which
whence
Vn = Z  Mn
Hv = m — STIf ;
SM
z = ,
n
nz = SAf,
nz  SM = 0;
2i) = 0, ...
(4)
which was to be proved.
In the case of the weighted arithmetic mean the sum of the weighted
residuals equals zero. This is proved as follows:
Let n = the number of observations;
Ml, M2, ,. . Mn = the observed values;
Pi, P2, ■ ■ ■ Pn = the corresponding weights;
z = the weighted arithmetic mean;
1^1, V2, . . . Vn = the residual errors;
then
but
or
pivi = pi(z  Ml)
P2V2 = P2(z  M2)
PnVn = Pn{Z — Mn)
Upv = Sp2  I,pM;
Z =
Spz = I,pM,
DEFINITIONS AND PEINCIPLES 247
from which
2pz  2pM =0;
whence
^pv =0, (5)
which was to be proved.
136. Sources of Error. The errors existing in observed values
may be due to mistakes, systematic errors, accidental errors, or
the least count of the instrument.
A mistake is, as its name implies, an error in reading or record
ing a result, and is not supposed to have escaped detection and
correction.
A systematic error is one that follows some definite law, and is
hence free from any element of chance. Errors of this kind may
be classed as atmospheric errors, such as the effect of refraction
on a vertical angle, or the effect of temperature on a steel tape;
instrumental errors, such as those 'due to index errors or imperfect
adjustments; and personal errors, such as individual peculiarities
in always reading a scale a little too small, or in recording a star
transit a little too late. Systematic errors usually affect all the
observations in the same manner, and thus tend to escape detec
tion by failing to appear as discrepancies. Such errors, however,
are in general well understood, and are supposed to be eliminated
by the method of observing or by subsequent reduction.
An accidental error is one that happens purely as a matter of
chance, and not in obedience to any fixed law. Thus, for instance,
in bisecting a target an observer will sometimes err a little to
the right, and sometimes a little to the left, without any assignable
cause; a steel tape will be slightly lengthened or shortened by a
momentary change of temperature due to a passing current of
air, and so on.
An error due to the least count of the instrument is one that is
caused by a measurement that is not capable of exact expression
in terms of the least count. Thus an angle may be read to the
nearest second by an instrument which has a least count of this
value, but the true value of the angle may differ from this reading
by some fraction of a second which can not be read.
137. Nature of Accidental Errors. Errors of this kind are
due to the limitations of the instruments used; the estimations
required in making bisections, scale readings, etc., and the con
248 GEODETIC SURVEYING
stantly changing conditions during the progress of an observa
tion. Each individual error is usually very minute, but the
possible number of such errors that may occur in any one measure
ment is almost without limit. In general it may be said that any
single observation is affected by a very large number of such errors,
the total accidental error being due to the algebraic sum of these
small individual errors. Thus in measuring a horizontal angle
with a transit the instrument is seldom in a perfectly stable posi
tion; the leveling is not perfect; the lines and levels of the instru
ment are affected by the wind and varying temperatures; the
graduations are not perfect; the reading is affected by the judg
ment of the observer; the target is bisected only by estimation;
the line of sight is subject to irregular sidewise refraction due to
changing air currents; and so on. As long as the component
errors are all accidental, however, the total error may be regarded
as a single accidental error.
133. The Laws of Chance. The errors remaining in ol)served
values after all possible corrections have been made are presumed
to be accidental errors, and must therefore be assumed to have
occurred in accordance with the laws of chance. By the laws of
chance are meant those laws which determine the probability of
occurrence of events which happen by chance.
By the ■probability of an event is meant the relative frequency
of its occurrence. It is not only a reasonable assumption but also
a matter of common experience, that in the long run the relative
frequency with which a proposed event occurs will closely approach
the relative possibilities of the case. Thus in tossing a coin
heads may come up as one possibility out of the two possibilities
of heads or tails, so that the probability of a head coming up is
onehalf; and in a very large number of trials the occurrence of
heads will closely approximate onehalf the total nvunber of trials.
Probabilities are therefore represented by fractions ranging in
value from zero to unity, in which zero represents impossibility of
occurrence, while unity represents certainty of occurrence.
The three fundamental laws of chance are those relating to
simple events, compound events, and concurrent events.
139. A Simple Event is one involving a single condition which
must be satisfied. The probability of a simple event is equal to
the relative possibility of its occurrence. Thus the probability of
drawing an ace from a pack of cards is iV, since there are four
DEFINITIONS AND PRINCIPLES 249
such possibilities out of 52, and f^ = xV; but the probability
of drawing an ace of clubs, for instance, is only jV, since there
is only one such possibility out of 52.
140. A Compound Event is one involving two or more con
ditions of which only one is required to be satisfied. The proba
bility of a compound event is equal to the sum of the probabilities of
the component simple events. This law is evidently true, since the
number of favorable possibilities for the compound event equals
the sum of the corresponding simple possibilities, and the total
number of possibilities remains unchanged. Thus the probability
of getting either a club or a spade in a single draw from a pack
of cards is onehalf, because the probability of getting a club is one
quarter, and the probability of getting a spade is onequarter, and
i  i = i; or in other words the 13 chances for getting a club
are added to the 13 chances. for getting a spade, making 26 favor
able possibilities out of a total of 52. The probability of draw
ing either a club, spade, heart, or diamond, equals i  i  i  i,
which equals unity, since the proposed event is a certainty.
141. A Concurrent Event is one involving two or more con
ditions, all of which are required to be satisfied together. The
probability of a concurrent event is equal to the product of the prob
abilities of the component simple events. This law is evidently
true, since the number of favorable possibilities for the concurrent
event is equal to the product of the corresponding simple pos
sibilities; while the total niraiber of possibilities is equal to the
product of the corresponding totals for the component simple
events. Thus the probability of cutting an ace in a pack of cards
is ^V, so, that the probability of getting two aces by cutting two
packs of cards is ^ X gV = *^ ^ /a ~ tV X yV = ttt It is evi
dent that the required condition will be satisfied if any one of the
four aces in one pack is matched with any one of the four aces in the
other pack, so that there are 4X4 favorable possibilities. Also
the cutting may result in getting any one of 52 cards in one pack
against any one of 52 cards in the other pack, so that there are
52X52 total possibilities. Multipljang the two probabihties,
therefore, gives the relative possibility and therefore the required
probability for the given concurrent event. Similarly the propo
sition may be proved for a concurrent event involving any
number of simple events. Thus in throwing three dice the
probability of getting 3 fours, for instance, will be XiXi=2Tir;
250 GEODETIC SUEVEYING
the probability of drawing a deuce from a pack of cards at
the same time that an ace is thrown with a die, will be
iVXt = tV; and so on.
In figuring the probability of a concurrent event it is neces
sary to guard against two possible sources of error. In the
first place the probabilities of the simple events involved in a
concurrent event may be changed by the concurrent condition.
Thus the probability of drawing a red card from a pack is f ,
but the probability of drawing two red cards in succession from
a pack is not MXfl, but ff Xt, since the drawing of the first
card changes the conditions under which the second card is drawn.
In the second place, the probability of a concurrent event may
be modified by the sense in which the order of simple events
may be involved. Thus in cutting two packs of cards the prob
ability that the first pack will cut an ace and the second a king
is tVXi\ = tb'5; but the probability that the first pack will cut
a king and the second an ace is also rTXTV = Ti¥; so that the
probability of cutting an ace and a king without regard to
specific packs becomes y^ti and not xb^t, as might be inferred.
142. Misapplication of the Laws of Chance. The probability
of a given event is the relative frequency of its occurrence in
the long run, and not in a limited number of cases. It is not
to be expected that every two tosses of a coin will result in one
head and one tail, since other arrangements are possible, and
the laws of chance are founded on the idea that every possible
event will occur its proportionate number of times. Thus in
the case of a coin we have for all possible events in two tosses.
Probability of 2 heads = J
" 1 head and 1 tail = 4
" 1 tail and 1 head = 
2 tails = i
Some one of these events must happen, so that the total prob
ability is i+i+4.+ i, which equals unity, as it should in a
case of certainty. The probability of two tosses including a
head and a tail (which may occur in two ways) is i+i=, so
that the proposed event is not one that occurs at every trial,
as is often inferred.
An event whose probability is extremely high will not neces
sarily happen on a given occasion, and this failure to happen
DEFINITIONS AND PEINOIPLES 251
does not imply an error in the theory of probabilities. The
very fact that the given probability is not quite unity indicates
the chance of occasional failures. Similarly an event with a
very small probability will sometimes happen, otherwise its
probability should be precisely and not approximately zero.
The probability of a future event is not affected by the
result of events which have already taken place. Thus if a tossed
coin has resulted in heads ten times in succession it is natural
to look on a new toss as much more likely to result in tails than
in heads; but mature thought will show that the probabilities are
still onehalf and onehalf for any new toss that may be made. The
confusion in, such a case comes from regarding the ten successive
heads as an abnormal occurrence, whereas, being one of the
possible occurrences, it should happen in due course along with
all other possible events. If tails were more likely to come up
than heads in any particular toss, it would imply some difference
of conditions instead of any overlapping influence. If the toss
of a coin is ever regarded as a matter of chance, it must always
be so regarded.
CHAPTER X
THE THEORY OF ERRORS
143. The Laws of Accidental Error. The mathematical
theory of errors relates entirely to those errors which are purely
accidental, and which therefore follow the laws of probability.
Mistakes or blunders, which follow no law, and systematic
errors, which follow special laws for each individual case, can
not be included in. such a discussion. If a sufficient number of
observations are taken it is found by experience that the accidental
errors which occur in the results are governed by the four fol
lowing laws :
1. Plus and minus errors of the same magnitude occur with
equal frequency.
This law is a necessary consequence of the accidental char
acter of the errors. An excess of plus or minus errors would
indicate some cause favoring that condition, whereas only acci
dental errors are under consideration.
2. Errors of increasing magnitude occur loith decreasing frequency.
This law is the result of experience, but for mathematical
purposes it is replaced by the equivalent statement that errors
of increasing magnitude occur with decreasing facility. For
reasons yet to appear (Art. 146) the facility of an error is rated
in units that make it proportional to the relative frequency with
which that error occurs instead of equal thereto.
3. Very large errors do not occur at all.
This law is also the result of experience, but it is not in
suitable form for mathematical expression. It is satisfactorily
replaced by the assumption that very large errors occur with
great infrequency.
4. Accidental errors are systematically modified by the cir
cumstances of observation.
This law is a necessary consequence of the first three laws,
and emphasizes the fact that these three laws always hold good
252
THE THEORY OF ERROES
253
however much the absolute values of the errors may be modified
by favorable or unfavorable conditions. The chief circumstances
affecting a set of observations are the atmospheric conditions,
the skill of the observer, and the precision of the instruments.
144. Graphical Representation of the Laws of Error. The
four laws of error are graphically represented in Fig. 63, in which
the solid curve corresponds to a series of observations taken
under a certain set of conditions, and the dotted curve to a
senes of observations taken under more favorable conditions.
For reasons which will appear in due course any such curve is
called a probability curve. The line XX, or axis of x, is taken
as the axis of errors, and the line AY, or axis of y, as the axis
of facility, the point A being taken as the origin of coordinates.
Thus in the case of the solid curve, if the line Aa represents any
,^''
'^•^
<\.
/
I
\.
^^•^y
^
'^V
^ — "^^r^^
"e
^
:::=^^^^^^^''
A a d
Pig. 63. — Probability Curves.
proposed error, then the ordinate db represents the facility with
which that error occurs in the case assimied. The first law is
illustrated by making the curves symmetrical with reference to
the axis of y, so that the ordinates are equal for corresponding
plus and minus values of x. The second law is illustrated by the
decreasing ordinates as the plus and minus abscissas are increased
in length. The third law does not admit of exact representation,
since a mathematical curve can not have all its ordinates equal
to zero after passing a certain point; a satisfactory result is
reached, however, by making all ordinates after a certain point
extremely small, with the axis of x as an asymptote to the curves.
The fourth law is illustrated by means of the solid curve and the
dotted curve, both of which are consistent with the first three
laws, but which have different ordinates for the same proposed
error. Thus small errors, such as Aa, occur with greater frequency
(or greater facility) in the case of the dotted curve than in the
254 GEODETIC SURVEYING
case of the solid curve, as shown by the ordinate ac being longer
than the ordinate ab\ while large errors, such as Ad, occur with
less frequency (or less facility) in the case of the dotted curve
than in the case of the solid curve, as shown by the ordinate
de being shorter than the ordinate d/.
145. The Two Tjrpes of Error. The recorded readings in
any series of observations are subject to two distinct types of
error. The first type of error includes all those errors involved
in the making of the measurement, such as those due to imper
fect instrumental adjustments, unfavorable atmospheric conditions,
imperfect bisection of targets, imperfect estimation of scale
readings, etc. The second type of error is that involved in
the reading or recording of the result, which must be done in
terms of some definite least count which excludes all inter
mediate values.
A given reading, therefore, does not indicate that precisely that
value has been reached in the process of measurement, but only
such a value as must be represented by that reading; so that
a given reading may be due to any one of an infinite number
of possible values lying within the limits of the least count.
Similarly, the error in the recorded reading does not indicate
that precisely that error has been made in the process of measure
ment, but only such an error as must be represented by that
value; so that the error of the recorded reading may in fact
be due to any one of an infinite number of possible errors Ijang
within the limits of the least count. The first type of error
is the true type or that which corresponds to the accidental
conditions under which a series of observations are made, while
the second type is a false type or definite condition or limitation
under which the work must be done. Thus in sighting at a target
a nimiber of times the angular errors of bisection may vary
among themselves by amounts which can only be expressed in
indefinitely small decimals of a second. If the least count
recognized in recording the scale readings is one second, however,
the recorded readings and the corresponding errors will vary among
themselves by amounts which differ by even seconds. The
probability curve of the preceding article is based on the first
type of error only, and is therefore a mathematically con
tinuous curve, since all values of the error are possible with this
type. In speaking of the errors of observations, however, the
THE THEORY OF ERRORS 265
errors of the recorded values are in general understood, and these
must necessarily differ among themselves by exactly the value
of the least count.
146. The Facility of Error. If an instrument is correctly
read to any given least count, no reading can be in error by more
than plus or minus a half of this least count; or, in other words,
each reading is the central value of an infinite number of
possible values lying within the limits of the least count. If
a great many observations are taken on a given magnitude, each
particular reading will be found to repeat itself with more or
less frequency, since all values lying within a half of the least
count of that particular reading must be recorded with the
value of that reading. If the same instrument, however, carried
finer graduations, with the least count half the previous value,
each reading would represent only those values within half
the previous limits. There would then be twice as many repre
sentative readings, with each one standing for half as many
actual values as with the coarser graduations. It is thus seen
that the relative frequency with which a given reading (and
the corresponding error) occurs, is directly proportional to the
least count of the instrmnent, or least count used in recording
the readings. Just as each reading is taken to represent an
infinite number of possible values within the limits of the least
count, so that reading must correspond to an infinite number of
possible errors within the same limits, each possible error having
a different facility of occurrence. Since in the long run, however,
each reading will be practically the average of all the values
that it represents, so the facility of the error due to that reading
may be taken practically as the average facility of all the corre
sponding errors. By definition (Art. 143) the facility of a given
accidental error is proportional to the frequency of its occurrence.
It is thus seen that the relative frequency with which a given
error (representing all possible errors due to a given reading)
occurs, is proportional to the facility of that error. Since the
relative frequency with which a given error occurs is proportional
to both its facility and the least count, it is proportional to
t^heir product, and is always made equal to this product by using
a suitable scale of facility. The facility of a given error is hence
equal to the relative frequency of occurrence of that error divided
by the least count.
256
GEODETIC SURVEYING
147. The Probability of Error. By the prohdbility of an
error is meant the relative frequency of its occurrence. Thus
in the measurement of an angle, if a given error occurred (on
the average) 27 times in 1000 observations, then the probability
that an additional measurement would be in error by that same
amount would be xHir The probability of a given error being
identical with its relative frequency of occurrence is hence (Art.
146) equal to the product of the facility of that error by the least
count. The probabihty of error for a certain set of conditions
is illustrated in Fig. 64. In this figure the spaces da, ae, eb, and
bf are each equal to onehalf of the least count. The probability
that an error Aa will occur is hence, in accordance with the
above principles, equal to the product of am (the facility) by
>
f
s
\
»
7
l\
__^^^^^
1
1
^v
"r__
A d a e h f a
Fig. 64. — The Trobability of Error.
de (the least count). As the least count is always very small,
we may write without appreciable error,
Probability of error Aa = amXde = area dste.
But (Art. 145) the error .4 a in the recorded reading includes all
the possible errors lying between Ad and Ae, that is, within
half the least count each way from Aa. The area dste therefore
represents the probability that the actual error committed lies
between the values Ad and Ae. Similarly the area etuf represents
the probability of an actual error between the values Ae and
Af. The probability that an actual error shall lie either between
Ad and .4e or between Ae and Af (compound event. Art. 140),
or in other words between Ad and Af, is equal to the sum of the
two separate probabilities, that is, to the combined area dsuf.
Or, in general, the probability that an error shall fall between
any two values Ac and Ag, is represented by the area included
between the corresponding ordinates cr and gv. On account
THE THEORY OF ERRORS 257
of this characteristic property the curve of facilities is commonly
called the probability curve. Strictly speaking the ordinates
limiting the area can only occur at certain equally spaced intervals
depending on the least count, but no material error is ever intro
duced by drawing them at any points whatever.
148. The Law of the Facility of Error is that law which con
nects all the possible errors in any set of observations with their
corresponding facilities, and is expressed analytically by the
equation of the probability curve. The law which governs the
occurrence of errors in any particular set of observations is
necessarily unknown and beyond determination, being the com
bined result of an uncertain number of variable and unknown
causes. Fortunately, however, it is found by experience that
there is one particular form of law which (with proper constants)
very closely represents the facility of error in all classes of obser
vations. This form of law is that which is in accordance with
the assiunption that the arithmetic mean of the observed values
is the most probable value when the same magnitude has been
observed a large number of times under the same conditions.
The same form of law being accepted as satisfactory in all cases,
therefore, the law for any particular case is determined by the
substitution of the proper constants.
149. Form of the Probability Equation. If x represents any
possible error and y the facility of its occurrence, we may write
y = 4>ix), (6)
which is read y equals a function of x. When the form of this
fimction has been determined the expression will be the general
equation of the probability curve. Since the probability that
the error x (of a recorded reading) will occur is equal (Art. 147)
to its facility multiplied by the least count, we have
P = yJx = ^{x)Jx, (7)
in which P is the probability of the occurrence of the error x,
and dx is the least count. If xi, X2, . . . Xn are the true errors in
the observed values of any magnitude Z, and Pi, P2, . . . Pn
are the corresponding probabilities of occurrence, we thus have
Pi = (j}(xi)Jx, P2 = (I>{x2)dx, etc.
258 GEODETIC SURVEYING
The probability P of the occurrence of this particular series
of errors, xi, X2, etc., in a set of observations of equal weight,
being a concurrent event (Art. 141), is equal to the product
of the individual probabilities, giving
P =4>{x{){x2)...4>{xn){^xy; .... (8)
whence
log P = log 4>{xi) + log 4>^X2). . . + log {vi) ^ dl0g4>{V2) _^ d log 4>{Vn) ^ Q ^Q,
dvi dv2 ' ' ' dv„ '
which may be written
/ d log ^(.0 \ ^ ^ / d log j^fe) \_^/ d log ^(.„) \
\ VidVi / V V2dV2 J \ VndVn )
But it has already been decided (Art. 134) that the arithmetic
mean of such a series of observed values is the most probable
value of the quantity observed. The adoption of the arith
metic mean as the most probable value, however, requires
the algebraic sum of the residuals (Art. 135) to reduce to zero;
whence
t)i + ■U2 . . . + v« = (12)
Since v\, V2, etc., are the result of chance, and hence independent
of each other, it follows from Eq. (12) that the coefficients of
Vi, etc., in Eq. (11) must all have the same value. Representing
this unknown value for any particular set of observations by the
THE THEOEY OF EEEOES 259
constant k, we have as the general condition which makes the
arithmetic mean the most probable value,
vdv '
whence by transposition
d log (f> (v) = kvdv.
Integrating this equation
log iv) = ce**"', (13)
in which e equals the base of the Naperian system of logarithms.
It is necessary at this point to remember that the probability
of the occurrence of a given error does not involve the question
as to whether we are right or wrong in assuming that an error of
that value has occurred in a particular observation. Thus in
the preceding discussion the probabilities assigned to the assumed
values of vi, V2, etc., are the probabilities for true errors of these
values, regardless of whether such errors have or have not occurred
in the given case. It is of the utmost importance, therefore, to
realize that Eq. (13) is not based on the assmnption that the
error v has occurred, but is ^ general statement of fact concern
ing any true error whose magnitude is v. Replacing v in Eq. (13)
by X, the adopted symbol for true errors, we have
but from equation (6)
whence
^{x) = ce*'^^';
y =^(x);
y = ce^kx\
260 GEODETIC SURVEYING
Since the facility y decreases as the numerical value of x increases,
it follows that \k is, essentially negative, and it is therefore
commonly replaced by — hj^. Making this substitution, we have
V = ce^"'\ (14)
in which y equals the facility with which any error x occurs,
c and h are unknown constants depending on the circumstances
of observation, and e is the base of the Naperian system of log
arithms. Though correct in apparent form, Eq. (14) must not
yet be regarded as the general equation of the probability curve,
since the quantities c and h appear as arbitrary constants,
whereas t wi 1 be shown in the next article that these values are
dependent on each other.
150. General Equation of the Probability Curve. The proba
bility that an error shall fall between any two given values
(Art. 147) is equal to the area between the corresponding ordi
nates of the probability curve. The probability that an error shall
fall between — oo and  oo is therefore equal to the entire area
of the curve. But it is absolutely certain that any error which
may occur will fall between these extreme limits, and the proba
bility of a certain event (Art. 138) is equal to unity. The entire
area of any curve represented by Eq. (14) must therefore be equal
to unity. Since all probability curves have the same total area,
it follows that any change in h will require a compensating change
in c; or, in other words,''c must be a function of h. The general
expression for the area of any plane curve is
=jydx
Substituting the value of y from Eq. (14)
The probability P that an error x will fall between the limits a
and h, is therefore
P
= f cef^'^'dx, (15)
THE THEORY OF ERRORS 261
and between the limits — oo and + oo , is
P =f cef^'^'dx =cf"^ ef^'^'dx.
But this probability, being a certainty, equals unity; whence
«' — 00
or
/:
The second member of this equation is a definite integral whose
evaluation by the methods of the calculus (for which such works
should be seen) gives
£
hence
c ~ h '
and
h
which substituted in Eq. (15) gives for the probabiUty P that an
error x will fall between any limits a and b,
P =^(V^''^'dx (16)
Also substituting the above value of c in Eq. (14) we have for the
general equation of the probability curve
y=^e^^^\ (17)
in which y is the facility with, which any error x occurs, e
( = 2.7182818) is the base of the Naperian system of logarithms,
and h (called the precision factor) is a constant depending on the
circumstances of observation. The constant h is the only element
262 GEODETIC SUEVBYING
in Eq. (17) which can vary with the precision of the work, and
therefore of necessity becomes the measure of that precision.
151. The Value of the Precision Factor. The general equa
tion of the probability curve is given by Eq. (17), but the definite
equation for any particular set of observations is not known
until the corresponding value of h has been determined. The
probability that an error x will occur (Art. 149) is
P = yJx = ^(x)Jx.
Substituting the value of y from Eq. (17),
P =^e^''''Jx =^{x)Jx (18)
With an infinite number of observations any residual v^ would be
infinitely close to the corresponding true error xi, and the relative
frequency with which vi occurred would not differ appreciably
from Pi. The value of h for any particular case could thus be
found from Eq. (18) by substituting these values for P and x.
As the number of observations is always limited, however, the
best that can be done is to find the most probable value of h
for the given case. The probability that a given set of errors
has occurred is, by Eq. (8),
P = {X2). . . 4>{Xn) • (ix)".
But from Eqs. (6) and (17)
so that
and
{x{) = — = e'''^>', etc.;
P = (^ye'''^^'(ia;)«,
7t
log P ^ n log h — h^Hx^ + n log Ax — ^Tog ;:;
whence by making the first derivative with respect to h equal to
zero
^ 2llx^h =0.
n
THE THEOEY OF EERORS 263
Solving for h we have
...... (19)
 P^
"^223
2Sa;2 ' •
in which n is the number of observations taken, and Sa^ is the
sum of the squares of the true errors which have occurred. The
true errors, however, can never be known, and formula (19) must
therefore be modified so as to give the most probable value of h
that can be determined from the residual errors. A discussion
of this condition is beyond the scope of this book, but for observa
tions of equal (or unit) weight results in the formula
^=yl2^' (20)
in which n as before is the number of observations that have been
taken, and HiP is the sum of the squares of the residual errors.
For observations of unequal weight (Art. 133) formula (19)
becomes
'4
2^pv^''
(21)
in which Spy^ is the sum of the weighted squares of the residuals,
and h as before is the precision factor for observations of unit
weight.
For the general case of indirect observations (Art. 168) on inde
pendent quantities, that is, with no conditional equations (Art. 131),
formula (19) becomes
^=yli
2Spi;2
(22)
in which n is the number of observation equations, q is the number
of imknown quantities, ^piP is the sum of the weighted squares
of the residuals, and h is the precision factor for observations
of unit weight.
For the general case of indirect observations involving con
ditional equations, formula (19) becomes
h = ^^ ovLI " ' (23)
jn — q + c
2J:piP
264
GEODETIC SURVEYING
in which c is the number of conditional equations, n is the number
of observation equations, q is the number of unknown quantities,
Spy2 is the sum of the weighted squares of the residuals, and h is
the precision factor for observations of unit weight. As will be
understood later (Art. 166), the number of independent unknomis
is always reduced by an amount which equals the number of
conditional equations, so that q in Eq. (22) is simply replaced by
(q  c) in Eq. (23).
152. Comparison of Theory and Experience. In the Funda
menta Astronomice Bessel gives the following comparison of theory
and experience. In a series of 470 observations by Bradley on
the right ascensions of Sirius and Altair the value of h was found
to be 1.80865, giving rise to the following table:
Probability of
Errors.
Number of Errors
By Theory.
By Experience.
0.0
0.1
0.2018
94.8
94
0.1
0.2
0.1889
88.8
88
0.2
0.3
0.1666
78.3
78
0.3
0.4
0.1364
64.1
58
0.4
0.5
0.1053
49.5
51
0.5
0.6
0.0761
35.8
36
0.6
0.7
0.0514
24.2
26
0.7
0.8
0.0328
15.4
14
0.8
0.9
0.0194
9.1
10
0.9
1.0
0.0107
5.0
7
1.0
00
0.0106
5.0
8
Totals 1.0000
470.0
470
The last column in this table tacitly assumes that the true errors
do not differ materially from the residual errors, the true errors
being of course unknown. The agreement of theory and expe
rience is very satisfactory.
There are two important points to be observed in applying
the theory of errors to the results obtained in practical work.
THE THEOEY OE EERORS 265
In the first place, the theory of errors presupposes that a very large
number of observations have been made. It is customary, how
ever, to apply the theory to any number of observations, however
limited. It is evident in such cases that reasonable judgment
must be used in interpreting the results obtained by the applica
tion of the theory. In the second place, the theory of errors is
the theory of accidental errors. It is in general impossible to
entirely prevent systematic errors in a process of observation;
and such errors can not be discovered or eliminated by any num
ber of observations, however great, if the circumstances of observa
tion remain unchanged. The theory of errors, therefore, makes
no pretense of discovering the truth in any case, but only to
determine the best conclusions that can be drawn from the observa
tions that have been made.
CHAPTER XI
MOST PROBABLE VALUES OE INDEPENDENT QUANTITIES
153. General Considerations. In accordance with the dis
cussions of the previous chapter it is evident that the true value
of an observed quantity can never be found. Adopting any
particular value for the observed quantity is equivalent to assum
ing that a certain series of errors has occurred in the observed
values. Manifestly the most probable value of the observed
quantity is that which corresponds to the most probable series
of errors; or, in other words, that series of errors which has the
highest probability of occurrence. It is therefore by means of
the theory of errors (Chapter X) that rules are established for
determining the most probable values of observed quantities.
154. Fundamental Principle of Least Squares. For the general
equation of the probability curve, Eq. (17), Art. 150, we have
Vtt
in which y is the facility of occurrence of any error x under the
conditions represented by the precision factor h. The probability
that any error x will occur (Art. 147) is equal to its facility multi
plied by the least count, or
P = yAx.
Hence if x^, X2, . ■ ■ Xn are the errors in the observed values
of any magnitude Z, and Pi, P2, . . . P„ are the corre
sponding probabilities of occurrence, we have
h h
2/1 =^:^e'''^'^ 2/2 = ^= e~'''^^', etc.,
Vtt Vtt
and
Pi = yiJx, P2 = 2/2^^, etc.
266
PROBABLE VALUES OF INDEPENDENT QUANTITIES 267
The probability P of the occurrence of this particular series of
errors Xi, x^, etc., in the given set of observations, being a con
current event (Art. 141), is equal to the product of the individual
probabilities, giving
P = (2/12/2 . . . VnWxY = i^j^^ e^'^^\AxY.
This equation is true for any proposed series of errors, and
hence for that series of residual errors t'j, vi, . . . Vn, which
results from assigning the most probable value to the observed
quantity. In this case Sx^ becomes Sw^, and we have
P = (^''e'''^^'{Ax)« (24)
But (Art. 153) the most probable value of the observed quantity
corresponds to that series of errors which has the highest prob
ability of occurrence. The most probable value z of any observed
quantity Z, therefore, requires P in Eq. (24) to be a maximum,
and this in turn requires l!,v^ to be a minimum. We thus have the
following
Pkinciple: In observations of equal precision the most probable
values of the observed quantities are those that render the sum of the
squares of the residual errors a minimum.
It is on account of this principle that the Method of Least
Squares has been so named.
155. Direct Observations of Equal Weight. A direct observa
tion (Art. 133) is one that is made directly on the quantity whose
value is to be determined. When the given magnitude is measured
a number of times under the same conditions (as represented
by the same precision factor h in the probability curve), the results
obtained are said to be of equal weight or precision. In such a case
the most probable value of the quantity sought mu^t accord with
the principle of the previous article, that is, the sum of the squares
of the residual errors must be a minimum.
Let z = the most probable value of a given magnitude;
n = the number of measurements taken;
Ml, M% . . . Mn = the several measured values;
then (Art. 154)
(Ml  z)^ + (Ma  s)2 . . . + (M„  2)2 = a minimum.
268 GEODETIC SURVEYING
Placing the first derivative equal to zero,
2(M, z) +2{M2z)... + 2{M„  z) =0;
whence
(Ml + Mg . . . + M„) nz = 0,
and
Ml + Ms . . . + M„ I:M
(25)
or, expressed in words, in the case of direct observations of equal
weight the most probable value of the unknown quantity is equal
to the arithmetic mean of the observed values. The above
discussion, however, must not be regarded as a proof of this
principle of the arithmetic mean, since (Art. 149) this very prin
ciple was one of the conditions under which the equation of
the probability curve was deduced. Eq. (25) therefore simply
shows that the equation of the probability curve is correct in form
and consistent with this principle.
Example. The observed values (of equal weight) of an angle A are
29° 21' 59".l, 29° 22' 06".4, and 29° 21' 58".l. What is the most probable
value?
29° 21' 59".l
29 22 06 .4
29 21 58 .1
3 )88 06 03 .6
29 22 01 .2
The most probable value is therefore 29° 22' 01".2.
156. General Principle of Least Squares. When' a given
magnitude is measured a number of times under different con
ditions (so that the precision factor corresponding to some of the
observations is not the same for all of them) , the results obtained
are said to be of unequal weight or precision. In accordance with
the sense in which weights are understood (Art. 133), an observa
tion assigned a weight of two means it is considered as good a
determination as the arithmetic mean of two observations of
unit weight, and so on. It is immaterial whether any one of the
observed values is considered of unit weight, as this is merely a
basis of comparison.
PROBABLE VALUES OF INDEPENDENT QUANTITIES 269
Let 2 = the most probable value of a given magni
tude;
Ml, M2, etc. = the values of the several measurements;
Pi, P2, etc. = the respective weights of these measure
ments;
api, ap2, etc. = the corresponding integral weights due to
the use of the factor a;
mi', mi", etc. = the api unit weight components of Mi
when considered as an arithmetical
mean;
W2', m2", etc. = similarly for M2, and so on;
vi, V2, etc. = the residuals due to Mi, M2, etc.;
then, as in Art. 134, we have
^ ^ mi' + mi" . . . __ Umi
^ api ~ api '
, , mi' + mi" . . . S?re2
M2 = ■ = ,
ap2 api
Sm _ ^ap.M _ S£M
Sap ~ Sap ~ Sp ^^^'
The value of z thus obtained is evidently independent of any
particular set of values that may be assigned to the components
mi', mi", etc., the components m2', m2", etc., and so on. Since
these various components are all of equal weight we must have
in accordance with Art. 154,
S(2; — mi)2 + S(z — m2)2 . . . + S(2— m„)2 = a minimum, (27)
as a criterion that must be satisfied when z is the most probable
value of the quantity Z. But, in accordance with Eq. (26),
this criterion must determine the same value of z no matter what
particular sets of values may be substituted for the components
mi', mi", etc., mi, mi' , etc., and so on. Adopting, therefore,
the particular sets of values
mi = mi" = .
, . . = Ml,
m2' = mi" = .
.. = Mi,
etc.
etc.,
270 GEODETIC SURVEYING
whence
^(z — mi)2 = api (z — MiY = apivi^,
2(z — m2)^ = ap2 (z — M^'^ = ap2V2,
etc. etc.,
and substituting in Eq. (27), we have
apiv^ + ap2V^. . . + apn'Vn = a minimum;
or, dividing out the common factor a,
PiVi^ + P2V2^ . . + PnV„^ = Si minimum. . . (28)
We thus have the following
General Pbinciple: In observations of unequal precision the
most probhble values of the observed quantities are those that render
the sum of the weighted squares of the residual errors a minimum.
157. Direct Observations of Unequal Weight. When a given
magnitude is directly measured a. number of times it may be
necessary to assign different weights to the results obtained, on
account of some change in the conditions governing the measure
ments. In such a case the most probable value of the quantity
sought must accord with the principle of the previous article,
that is, the sum of the weighted squares of the residual errors
must be a minimum.
Let z = the most probable value of a given magnitude;
Ml, M2, ■ . Mn = the several measured values;
Pi, p2, ■ ■ ■ Pn = the corresponding weights;
then (Art. 156)
Pi(M 1  zy + P2{M2  zy . . .+ PniMn — z)^ = 3, minimum.
Placing the first derivative equal to zero,
2pi(Mi  z) + 2p2'M2  2) ... + 2p„(M,  2) = 0;
PROBABLE VALUES OF INDEPENDENT QUANTITIES 271
whence
(pi Ml + P2M2 ...+ VnMn)  (Pi + P2 • ■ ■ + P«) 3 = 0,
and
Z _ Pl^l +P1M2. . . +VnMn ^ 2pM.
Vl + V2 . . . + Vn Hip ' ' ' ' ^ '
or, expressed in words, in the case of direct observations of unequal
weight the most probable value of the unknown quantity is equal
to the weighted arithmetic mean of the observed values. The
above discussion, however, must not be regarded as a proof of
this principle of the weighted arithmetic mean, since Eq. (29)
is deduced from a principle based in part on the truth of Eq. (26),
which is identical with Eq. (29). As the truth of Eq. (26) is
established in Art. 156, however, Eq. (29) shows that the general
principle of least squares leads to a correct result in a case where
the answer is already known.
Example. The observed values for the length of a certain base line are
4863.241 ft. (weight 2), and 4863.182 ft. (weight 1). What is the most
probable value?
4863.241 X 2 = 9726.482
4863.182 X 1 = 4863.182
3)14589.664
4863.221
The most probable value is therefore 4863.221 ft.
158. Indirect Observations. An indirect ohservation is one
that is made on some function of one or more quantities, instead
of being made directly on the quantities themselves. Thus in
measuring an angle by repetition the observation is indirect, as
the angle actually read is not the angle sought, but some multiple
thereof. Similarly when angles are measured in combination
the observations are indirect, since the values of the individual
angles must be deduced from the results obtained by some pro
cess of computation.
An observation equation is an equation expressing the function
observed and the value obtained. Thus if x, y, etc., represent
the unknown quantities whose values are to be deduced from the
272 GEODETIC SURVEYING
observation, we may have as observation equations such expres
sions as
6a; = 185° 19' 40",
or
7x + 102/  3z = 65.73,
according to the function observed.
In general the observation equations which occur in geodetic
work may be written in the following form :
aix + hiy + CiZ . . . = Mi (weight pi)
a^x + 622/ + C23 . . . = M2 (weight P2)
a„x + hnV + c„3 . . . = Mn (weight p„)
(30)
in which ai, 02, &i, ^2 etc., are known coefficients; x, y, etc., are
the unknown quantities; Mi, M2, etc., are the observed values;
and pi, p2, etc., are the respective weights of these values. If the
number of observation equations is less than the number of unknown
quantities, the values of x, y, z, etc., can not be found, nor even
their most probable values. If the number of observation equa
tions equals the nvunber of unknown quantities, the equations
may be solved as simultaneous equations, and each equation will
be exactly satisfied by the values obtained for x, y, z, etc., even
though these values are not the true values sought. If the num
ber of observation equations exceeds the number of unknown
quantities there will in general be no values of x, y, z, etc., which
will exactly satisfy all the equations, on account of the unavoidable
errors of observation. Hence if the most probable values of the
imknown quantities be substituted the equations will not be
exactly satisfied, but will reduce to small residuals vi, m, vz, etc.
If, therefore, x, y, z, etc., be understood to mean the most probable
values of these quantities, we will have
aiX + biy + CiZ . . . — ilf 1 = Vi (weight p{)
a^x + 622/ + C2Z . . . — M2 = «'2 (weight p^)
a„x + 6„2/ + c„z . . .  M„ = «;„ (weight p„)
(31)
PEOBAELE VALUES OF INDEPENDENT QUANTITIES 273
By a consideration of these equations, together with any special
conditions which must be satisfied, rules may be established for
finding the most probable values of the unknown quantities in
all cases of indirect observations.
159. Indirect Observations of Equal Weight on Independent
Quantities. An independent quantity is one whose value is
independent of the value of any other quantity under considera
tion. Thus in a line of levels the elevation of any particular
bench mark bears no necessary relation to the elevation of any
other bench mark; whereas in a triangle the three angles are not
independent of each other, as their sum must necessarily equal
180°.
In the case of indirect observations of equal weight on inde
pendent quantities, the most probable values of the tmknown
quantities are found by a direct application of the method of
normal equations. A normal equation is an equation of condi
tion which determines the most probable value of any one unknown
quantity corresponding to any particular set of values assigned to
the remaining unknowns. A normal equation must therefore
be specifically a normal equation in x, or in y, etc. By forming
a normal equation for each of the unknowns there will be as many
equations as unknown quantities. The solution of these equa
tions as simultaneous will give a set of values for the unknowns
in which each value is the most probable that is consistent with
the remaining values, which can only be the case when all the
values are simultaneously the most probable values of the unknown
quantities.
To establish a rule for forming the normal equations in the
case of equal weights let us rewrite Eqs. (31), omitting the
weights, thus:
aix + biy + ciz ... — Mi = Vi
0,2^ + b2y + C2Z . . . ~ M2  V2
o»a; + b„y + CnZ . . ■ — Mr, = K
(32)
In accordance with Art. 154 the most probable values of the
unknown quantities are those which give
vi^ + V2^ ■ ■ ■ + Vr? = & minimum.
274 GEODETIC SURVEYING
Since (in lormmg the normal equations) the most probable value
of X is desired for any assumed set of values for the remaining
unknowns, we place the first derivative with respect to x equal
to zero ; whence, omitting the common factor 2, we have
But from Eqs. (32), under the given assimiption of fixed values
for all quantities excepting x, we obtain
dvi dv2 ,
dx = """ 'dx = "'' "*°
whence by substitution,
aivi + 02^2 . ■ . + anVn = = normal equation in x.
In a similar manner we have
bivi + b2V2 . . ■ + b„Vn = = normal equation in y;
ciwi + 02^2 • • • + CnV„ = = normal equation in z;
etc., etc.;
and hence for forming the several normal equations in the case
of indirect observations of equal weight on independent quan
tities, we have the following
Rule : To form the normal equation for each one of the unknown
quantities, multiply each observation equation by the algebraic
coefficient of that unknown quantity in that equation, and add the
results.
Having formed the several normal equations, their solution
as simultaneous equations gives the most probable values of the
unknown quantities.
Examph 1. Given the observation equation
6x = 90° 15' 30"
In applying the above rule to this case we would have to multiply the whole
equation by 6, and then divide by 36 to obtain the most probable value
of X. It is evident that we would obtain the same value of x by dividing
the original equation by 6, so that in the case of a single equation with a
single unknown quantity the most probable value of that quantity is obtained
by simply solving the equation.
PROBABLE VALUES OF INDEPENDENT QUANTITIES 275
Example 2. Given the observation equations
2x = 124.72,
X = 62.31,
7x = 439.00.
Multiplying the first equation by 2, the second by 1, and the third by 7,
we have
4x = 249.44;
X = 62.31;
49x = 3073.00;
whence by addition we obtain the normal equation
54x= 3384.76,
the solution of which gives
X = 62.68,
which is hence the most probable value that can be obtained from the given
set of observations. The student is cautioned against adding up the obser
vation equations and solving for x, as this plan does not give the most
probable value in such cases.
Example 3. Given the observation equations
2x+ y = 31.65,
X  3i/ = 5.03,
X  y = 11.26.
Following the rule for normal equations, we have
ix + 2y = 63.30
X  3y = 5.03
X  y = 11.26
&x — 2y = 79.59 = normal equation in x;
and
2x + y = 31.65
 3x + 92/ =  15.09
 X + y =  1126
 2x \ lly = 5.30 = normal equation in y.
It is absolutely essential in forming the normal equations to multiply by
the algebraic coefficients as illustrated above, and not simply by the numerical
value of the coefficient. Bringing the normal equations together, we have
Ox  2y = 79.59,
 2x + 112/ = 5.30.
Attention is called to the fact that the coefficients in the first row and first
column are identical in sign, value, and order, and that the same is true of
the second row and second column. The same law would hold good if there
were a third row and a third column, and so on (Art. 162); and this is a
check that must never be neglected. Solving the two normal equations as
simultaneous equations, we have
X = 14.29 and y = 3.08,
and these are hence their most probable values.
276 GEODETIC SUEVEYING
160. Indirect Observations of Unequal Weight on Independent
Quantities. In the case of indirect observations of unequal
weight on independent quantities, the most probable values of
the unknown quantities are found by the solution of one or more
normal equations which involve the different weights in their
formation.
To establish a rule for forming the normal equations in the
case of unequal weights let us rewrite Eqs. (31), thus:
aix + biy + ciz . . . — Mi = vi (weight pi)
a2X + 62?/ + C2Z . . . — M2 = V2 (weight ^2)
anX + bny + CnZ ■ . .  Mn = Vn (weight Pn)
(33)
In accordance with Art. 156 the most probable values of the
imknown quantities are those which give
Piwi^ + P2«'2^ . . . + PnVr? = a minimum.
Since (in forming the normal equations, Art. 159) the most
probable value of x is desired for any assumed set of values for
the remaining imknowns, we place the first derivative with
respect to x equal to zero; whence, omitting the common
factor 2, we have
But from Eqs. (33), under the given assumption of fixed values
for all quantities excepting x, we obtain
dvi dv2
whence by substitution,
(aipi)vi + {a2P2)V2 . . . + {a„Pn)Vn = = normal equation in x.
In a similar manner we have
(bipijvi + (62P2)w2 . . . + {bnPn)Vn = = uormal equation in y;
(cipi)wi + {c2P2)V2 . . . + {cnPn)Vn = = normal equation in z;
etc., etc.;
PEOBABLE VALUES OF INDEPENDENT QUANTITIES 277
and hence for forming the several normal equations in the case
of indirect observations of unequal weight on independent quan
tities, we have the following
Rule : To form the normal equation for each one of the unknown
quantities, multiply each observation equation by the product of the
weight of that observation and the algebraic coefficient of that unknown
quantity in that equation, and add the results.
Having formed the several normal equations, their solution
as simultaneous equations gives the most probable values of the
unknown quantities.
Exampk 1 . Given the observation equations
3x = 15° 30' 34" .6 (weight 2),
5x = 25 50 55 .0 (weight 3).
Multiplying the first equation by 6 ( = 3 X'2), and the second equation by
15 ( = 5 X 3), we have
18x = 93°03'27".6;
75x = 387 43 45 .0;
whence by addition we obtain the normal equation
93x = 480° 47' 12".6,
the solution of which gives
x = 5° 10' ll".l,
which is hence the most probable value that can be obtained from the given
set of observations.
Example 2. Given the observation equations
x+ y = 10.90 (weight 3),
2x — y = 1.61 (weight 1),
X + 32/ = 24.49 (weight 2).
Following the rule for normal equations, we have
3x + 32/ = 32.70
4x  22/ = 3.22
2x + 62/ = 48.98
9x + 72/ = 84.90 = normal equation in x;
and
3x + 32/ = 32.70
2x + y = 1.61
6x + I82/ = 146.94
7x + 222/ = 178.03 = normal equation in y.
Solving these two normal equations as simultaneous, we have
X = 4.172, and y = 6.765,
and these are hence their most probable values.
278 GEODETIC SURVEYING
161. Reduction of Weighted Observations to Equivalent
Observations of Unit Weight. To establish a rule for this pur
pose let us rewrite Eqs. (30) , thus :
aix + biy + ciz . . . = Mi (weight pi),
a2X + hiV + C2Z . . . = M2 (weight P2) ,
a„x + bny + CnZ . . . = Mr, (weight p„).
Let C be such a factor as will change the first of these equations
to an equivalent equation of unit weight, so that we may write
Caix + Cbiy + Cciz . . . = CMi (weight 1),
a2X + 622/ + C2Z ■ ■ ■ = M2 (weight P2) ,
onX + bnV + c„z . . . = M„ (weight p„) ;
in which the most probable values of x, y, z, etc., are to remain the
same as in the original equations; or, in other words, the two
sets of equations are to lead to the same normal equations. In
accordance with the rule of Art. 160, we have from the first set
of equations
Normal
equation
in x
(piai^x+piaibiy+piaiciz . . . =piOiMi)
+ (p2a2^X+p2a2b2y + P2a2C2Z . . . =^202^2)
 ( etc., etc )
and from the second set of equations
(34)
Normal
equation
in X
(C^aiH+C^aibiy+C^aiciz . . . =C^aiMi)
+ ip2a2^x+p2a2b2y+p2a2C2Z . . . =p2a2M2)
+ ( etc., etc )
(35)
Comparing Eq. (34) with Eq. (35), term by term, we find they are
in all respects identical provided we write
whence
C = V^ (36)
PEOBABLE VALUES OF INDEPENDENT QUANTITIES 279
From the symmetry of the equations involved it is evident that
the same conclusion would result from a comparison of the nor
mal equations in y, z, etc. Hence it is seen that an observation
equation of any given weight may be reduced to an equivalent
equation of unit weight by multiplying the given equation by the
square root of the given weight. Evidently the converse of this
proposition is also true, so that an equation of unit weight can be
raised to an equivalent equation of any given weight by dividing
the given equation by the square root of the given weight. The
general laws of weights, as given in Art. 53, are readily derived
by an application of these two principles. The new equations
formed in the manner described, and taken in conjunction with
the new weights, may be used in any computations in place of the
original equations, whenever so desired.
Example 1. Given the observation equation
3a; = 8.66 (weight 4).
What is the equivalent observation equation of unit weight?
Since the square root of 4 is 2, we have
&x = 17.32 (weight 1)
as the equivalent equation.
Example 2. Given the observation equation
Zx + Qy = 11.04 (weight 1).
What is the equivalent observation equation of the weight 9?
Since the square root of 9 is 3, we have
s + 2)/ = 3.68 (weight 9)
as the equivalent equation.
Example 3. Given the observation equation
X + y — 2z = a (weight 3).
What is the equivalent observation equation of the weight 7?
Multiplying by ^3 and dividing by V?, we have
\/f a; + Vf V  2\/f z = Vf a (weight 7)
as the equivalent equation.
280 GEODETIC SURVEYING
162. Law of the Coefficients in Normal Equations. In accord
ance with Art. 158, we may write in general for any set of
observations
aix + hiy +ciz . . . = Mi (weight pi) ,
a2X + bzy + c^z . . . = M2 (weight P2) ,
arfl + hny + CnZ. . . = Mn (weight p„).
Forming the normal equation in x in accordance with the rule of
Art. 160, the multiplying factors are piai, ^2^2, etc., giving
piaiH + piaibiy + piaiaz . . . = piaiMi
P2a2^X + P20.2h2y + P2a2C2Z . . . = ^202^2
S(pa^)x+2i(pa6)2/ + S(pac)3 . . . =S(paM')= normal equation in x.
Similarly, for the normal equation in y, the multiplying factors
are pi6i, P2&2, etc., giving
T,{pah)x + 'S^{pb^)y + 'L{phc)z . . .=Il(p&M) = normal equation in j/.
Similarly, for the normal equation in z, the multiplying factors
are pici, P2C2, etc., giving
I!(pac)a;+S(p&c)?/ + 2(pc2)z . . .=S(pcikf) = normal equation in z ;
and so on for any additional unknown quantities. Collecting
the several normal equations together, we have
S(pa2)a; + ^{pah)y + ll{pac)z . . . = HipaM);
^{pab)x + S(p&2)2/ + '^{phc)z . . . = 2(p6ilf);
S(pac)a; + ll{phc)y + 2(pc2)z . . . = S(pcM);
etc., etc.
An examination of these equations shows that the coefficients in
the first row and in the first column are identical in sign, value,
and order. The same proposition is true of the second row and
second column, the third row and third column, and so on. This
is hence the general law of the coefficients in any set of normal
equations, and furnishes a check on the work that should never
be neglected.
PEOBABLE VALUES OF INDEPENDENT QUANTITIES 281
Example. Let the following observation equations be given:
2x  z = 8.71 (weight 2),
x2y + Zz = 2.16 (weight 1),
2/  2z = 1.07 (weight 2),
X Zy = 1.93 (weight 1).
I
[The corresponding normal equations are
IQx 
— 5a; 
— X 
from which we have
lOx — by — z = 38.93 = normal equation in x;
— 5x + 15y — lOz = — 7.97 = normal equation in y;
— X  lOy + 19z =  15.22 = normal equation in z;
„ ^ . , . / Fu:st row are + 10, — 5,1.
Coemcients m i ,,. , , , r. ^
I First column are +10,5,1.
„„.... / Second row are — 5, + 15, — 10.
(Joefncients m i c j i c ic ^n
L becond column are — 5, + 15, — 10.
ri a ■ i ■ / Third row are  1,  10, + 19.
Coefficients m [ ^^^^ ^^j^^^ ^^^  1,  loi + 19.
163. Reduced Observation Equations. Such observation equa
tions as are likely to occur in geodetic work may be written imder
the general form
ax + by + CZ + etc. = M (37)
Substituting
X = Xi \ Vi
y = yi + V2
Z = Zl + V3
(38)
in which xi, yi, 21, etc., are any assumed constants,, and vi, V2, V3,
etc., ate new imknowns, the equation takes the reduced form
avi + bv2 +CV3 + etc. = M — {axi + byi + czi + etc.). (39)
In this new equation it will be noticed that the first member is
identical in form with the first member of the original equation,
the only change being the substitution of the new variables for
the old ones; and that the second member is what the original
equation reduces to when the assumed constants are substituted
for the corresponding variables. The reduced observation
Eq. (39) may therefore be written out at once from the observa
282 GEODETIC SUEVEYING
tion Eq. (37), without going through the direct substitution of
Eqs. (38). Particular attention is called to the second member
of Eq. (39), in which it is seen that the result due in any case to
the use of the assumed values of x, y, etc., must always be sub
tracted from the corresponding measured value, and not vice
versa, as any error in sign will render the whole computation
worthless. It is also to be noted that the original weights apply
also to the reduced observation equations, since these are simply
different expressions for the original equations.
In view of the meaning of the terms in Eqs. (38) it is evident
that the most probable value of x is that which is due to the most
probable value of vi, and correspondingly with all the other
unknowns. We may, therefore, in any case, reduce all the
original observation equations to the form of Eq. (39), determine
from these reduced equations the most probable values of vi, V2,
etc., and then by means of Eqs. (38) determine the most probable
values of x, y, z, etc. The object of this method of computation
is to save labor by keeping all the work in small numbers. This
result is accomplished by assigning to xi, yi, etc., values which
are known to be approximately equal to x, y, etc., as this will
evidently reduce the second term of equations like Eq. (39) to
values approximating zero. Approximate values' of the unknowns
are always obtainable from an inspection of the observation,
equations, or by obvious combinations thereof.
Example 1. Given the following observation equations:
X = 178.651,
y = 204.196,
X + 2/ = 382.860,
2x + y = 561.522;
to find the most probable values of the unknowns by the method of reduced
observation equations.
Assuming for the most probable values
x = 178.651 + vi,
y = 204.196 + v^,
we have by substitution in the observation equations, or directly in accord
ance with Eq. (39),
«i = 0.000;
D2 = 0.000;
vi + Vi = 0.012;
2wi +V2 = 0.024.
PROBABLE VALUES OF INDEPENDENT QUANTITIES 283
Forming the normal equations from these reduced observation equations,
we have
6di + Swa = 0.060;
3!)i + 3v2 = 0.036;
whose solution gives
vi = 0.008 and V2 = 0.004;
whence for the most probable values of x and y we have
X = 178.651 + 0.008 = 178.659;
y = 204.196 + 0.004 = 204.200.
These results are identical with what would have been obtained if any other
values had been used for xi and yi, or if the normal equations had been
formed directly from the original observation equations.
Example 2. Given the following observation equations:
21 + 2/ = 116° 38' 19".7 (weight 2),
a; + 2/ = 73 17 22 .1 (weight 1),
X  y = 13 24 28 .3 (weight 3),
x + 2y = 103 13 47 .7 (weight 1);
to find the most probable values of the unknowns by the method of reduced
observation equations.
It is readily seen that the first two of these equations are exactly satisfied
if we write
X = 43° 20' 57".6;
2/ = 29 56 24 .5.
Adopting these as the approximate values we have for the most probable
values
a; = 43° 20' 67".6 + Vi;
y = 29 56 24 .5 + v^;
whence by substitution in the observation equations, or directly in accord
ance with Eq. (39), we have
2i;i + f 2 = 0".0 (weight 2);
vi + V2 = .0 (weight 1);
vi — «)2 = — 4 .8 (weight 3) ;
vi +2v2 = 1 .1 (weight 1).
Forming the normal equations from these reduced observation equations,
we have
132)1 + 4j)2 =  13".3;
4t;i + 10t>2 = 16 .6;
whose solution gives
vi 1".75 and «2 = + 2".36;
whence for the most probable values of x and y we have
X = (43° 20' 57".6)  1".75 = 43° 20' 55".86;
 y = (29 66 24 .5) + 2 .36 = 29 66 26 .86.
As in the previous example these results are identical with what would have
been obtained if any other values had been used for Xi and yi, or if the normal
equations had been formed directly from the original observation equations.
CHAPTER XII
MOST PROBABLE VALUES OF CONDITIONED AND COMPUTED
QUANTITIES
164. Conditional Equations. The methods heretofore given
determine the most probable values in all cases where the quanti
ties observed are independent of each other. In many cases, how
ever, certain rigorous conditions must also be satisfied, so that any
change in one quantity demands an equivalent change in one
or more other quantities. Thus in a triangle the three angles
can not have independent values, but only such values as will add
up to exactly 180°. When quantities are thus dependent on each
other they are called conditioned quantities. By an equation of
condition or a conditional equation is meant an equation which
expresses a relation that must exist among dependent quantities.
Thus if X, y, and z denote the three angles of a triangle we have
the corresponding conditional equation
x + y + z = 180°.
In such a case the most probable values of x, y, and z are not
those values which may be individually the most probable, but
those values which belong to the most probable set of values that
will satisfy the given conditional equation. In accordance with
the principles heretofore established that set of values is the most
probable which leads to a minimum value for the sum of the
weighted squares of the resulting residuals in the observation
equations.
In the problems which occur in geodetic work the conditional
equations may in general be expressed in the form
aix 1022/ ■ ■ ■ + aj = E^
bix + b2y . . . + bj = E^
mix + m2y . . . + mj, = E„
(40)
284
PROBABLE VALUES OF CONDITIONED QUANTITIES 285
in which x, y, t, etc., are the most probable values of the unknown
quantities, and u is the number of such quantities. It is evident
that the number of independent conditional equations must be
less than the number of unknown quantities. For if these equa
tions are equal in number with the unknown quantities their
solution as simultaneous equations will determine absolute values
for the unknowns, so that such quantities can not be the subject
of measurement. While if the number of these equations exceeds
the number of unknowns, such equations can not all be inde
pendent without some of them being inconsistent. On the other
hand the total mmiber of equations (sum of the observation and
the independent conditional equations) must exceed the number
of unknown quantities. For if the total number of equations is
equal to the number of unknown quantities, their solution as
simultaneous equations will furnish a set of values which will
exactly satisfy all the equations, without involving any question
of what values may be the most probable. While if the total
number of equations is less than the number of unknown
quantities the problem becomes indeterminate.
There are in general two methods of finding the most probable
values of the unknown quantities in cases involving conditioned
quantities. In the first method the conditional equations are
avoided (or eliminated) by impressing their significance on the 
observation equations, which reduces the problem to the cases
previously given. In the second method the observation equa
tions are eliminated by impressing their significance on the con
ditional equations, when the solution may be effected by the
method of correlatives (Art. 167). The first method is the most
direct in elementary problems, but the second method greatly
reduces the work ,of computation in the case of complicated
problems.
165. Avoidance of Conditional Equations. In a large num
ber of problems it is possible to avoid the use of conditional
equations by the manner in which the observation equations are
expressed. The conditions which have to be satisfied in any
given case are never alone sufficient to determine the values of
any of the unknown quantities, as otherwise these quantities
would not be the subject of observation. It is only after definite
values have been assigned to some of the unknown quantities
that the conditional equations limit the values of the remaining
286 GEODETIC SURVEYING
ones. In any problem, therefore, a certain number of values
may be regarded as independent of the conditional equations,
whence the remaining values become dependent on the independent
ones. Thus in a triangle any two of the angles may be regarded
as independent, whence the remaining one becomes dependent
on these two, since the total sum must be 180°. In any elementary
problem it is generally self evident as to how many quantities
must be regarded as independent, and which ones may be so taken.
In such cases the conditional equations may be avoided by
writing out all of the observation equations in terms of the
independent quantities. The most probable values of these
quantities may then be found by the regular rules for independent
quantities, whence the most probable values of the remaining
quantities are determined by the surrounding conditions that
must be satisfied.
Example 1. Referring to Fig. 65, the following angular measurements
have been made;
X = 28° 11' 52".2;
y = 30 42 22 .7;
z = 58 64 17 .6.
What are the most probable values of these angles?
It is evident from the figure that these angles are sub
ject to the condition
X + y = z.
If, however, we write the observation equations in the
form
X = 28° 11' 52".2;
y = 30 42 22 .7;
x+y = 58 54 17 .6;
the conditional equation is avoided, since x and y are manifestly inde
pendent angles. The second set of observation equations must lead to
exactly the same figures for the most probable values of x and y (and hence
for z) as the first set, since it is only another way of stating exactly the
same thing. Since x and y are independent angles we may write for the
most probable values
X = 28° 11' 52".2 + vi;
y = 30 42. 22 .7 + V2;
whence the reduced observation equations are
vi = 0".0;
!)2 = .0;
vi + vt = 2 .7.
Fig. 65.
PROBABLE VALUES OF CONDITIONED QUANTITIES 287
The corresponding normal equations are
2wi + V2
Vi + 2t)2
2".7;
2 .7;
whose solution gives
vi= + 0".9 and 1^2=+ 0".9.
The most probable values of the given angles are therefore
X = 28° 11' 53".l;
2/ = 30 42 23 .6;
z = 58 54 16 .7.
Example S. Referring to Fig. 66, the following angular measurements
have been made:
X = 80° 46' 37".6 (weight 2);
y = 135 08 14 .9 (weight 1);
z = 144 06 10 .8 (weight 3).
What are the most probable values of these angles?
It is evident from the figure that these angles are
subject to the condition
x + y + z = 360°.
Any two angles at a point, such as x and y, may
be regarded as independent, so that the conditional
equation is avoided by writing all the observation
equations in terms of these two quantities. Thus we
write:
X = 80° 45' 37".6 (weight 2);
y = 135 08 14 .9 (weight 1);
360°  (a; + 2/) = 144 06 10 .8 (weight 3) ;
whence by substituting
we have
X = 80° 45' 37".6 + vi,
y = 135 08 14 .9 + V2,
Vi = 0".0 (weight 2);
!)2 = .0 (weight 1);
t/i + f 2 = — 3 .3 (weight 3) ;
from which the normal equations are
5«i + Zv2=  9".9;
3!)i \ivi 9 .9;
whose solution gives
vi=  0".9 and v^ =  1".8.
The most probable values of the given angles are therefore
X = 80° 45' 36".7;
y = 135 08 13 .1;
2 = 144 06 10 .2.
288 GEODETIC SUEVEYING
166. Elimination of Conditional Equations. If the con
ditional equations can not be directly avoided, as in the
preceding article, the same result may be indirectly accomplished
by algebraic elimination, as about to be explained. The number
of unknown quantities (Art. 164) necessarily exceeds the number'
of independent conditional equations. The number of dependent
unknowns, however, can not exceed the number of independent
conditional equations, since any values whatever may be assigned
to the remaining unknowns and still leave the equations capable
of solution. Thus if there are five unknowns and three independent
conditional equations, any values may be assigned to any two of
the unknowns, leaving three equations with three imknowns and
hence capable of solution. The unknowns selected as arbitrary
values thus become independent quantities on which all the others
must depend, and the nmnber of unknowns which may be thus
selected as independent quantities is evidently equal to the
difference between the total number of unknowns and the number
of independent conditional equations. If the most probable
values are assigned to the independent quantities, the most
probable values of the dependent quantities then become known
by substituting the values of the independent quantities in the
dependent equations. The general plan of procedure is as
follows:
1. Determine the nmnber of independent unknowns by sub
tracting the number of conditional equations from the number
of unknown quantities.
2. Select this number of unknowns as independent quantities.
3. Transpose the conditional equations so that the dependent
quantities are all on the lefthand side and the independent quan
tities on the righthand side.
4. Solve the conditional equations for the dependent unknowns,
which will thus express each of these dependent unknowns in
terms of the independent unknowns.
5. Substitute these values of the dependent unknowns in the
observation equations, which will then contain nothing but
independent imknowns.
6. Find the most probable values of the independent unknowns
from these modified observation equations by the regular rules
for independent quantities.
7. Substitute these values of the independent unknowns in
PEOBABLE VALUES OF CONDITIONED QUANTITIES 289
the expressions for the dependent unknowns, alid thus determine
the most probable values of the remaining quantities.
Example. Given the following data, to find the most probable values
of X, y, and «:
[ X = 17.82 (weight 2);
• y = 15.11 (weight 4);
z = 29.16 (weight 3).
Observation equations ] y = 15.11 (weight 4);
Conditional equations {Hfl ^ ^Z'lS.
Th,e solution is as follows:
Number of observation equations = 3.
Number of conditional equations = 2.
Number of independent quantities = 1.
Let X be the independent quantity, and y and z the dependent quantities.
Transpose the conditional equations so as to leave only the dependent
quantities on the left hand side, thus:
5y = 112.00  2x;
y  z = 39.00  3a;.
Solve for the dependent quantities, giving the dependent equations
y = 22.40  0.4x;
z =  16.60 + 2.6a;.
Substitute in the observation equations, giving
X = 17.82 (weight 2);
22.40  0.4x = 15.11 (weight 4);
 16.60 + 2.6x = 29.16 (weight 3);
whence
X = 17.82 (weight 2);
0.4x = 7.29 (weight 4);
2.6x = 45.76 (weights);
in which x is an independent unknown. Forming the normal equation
by multiplying the above equations respectively by 2, 1.6, and 7.8, we have
2.00X = 35.640,
0.64x = 11.664,
20.28X = 356.928
22.92X = 404.232;
X = 17.637;
which, substituted in the first dependent equation, gives,
y = 22.40  0.4(17.637) = 15.345,
290 GEODETIC SURVEYING
and substituted in the second dependent equation, gives
z =  16.60 + 2.6(17.637) = 29.255;
so that for the most probable values of the unknown quantities, we have
X = 17.637;
y = 15.345;
z = 29.255.
As a check on the work of computation, we may substitute these values in
the conditional equations, giving
2x + 5y = 35.274 + 76.725 = 111.999;
3x+ y  z = 52.911 + 15.345  29.255 = 39.001;
from which it is seen that each equation checks with the corresponding
conditional equation within 0.001, which is an entirely satisfactory check.
The essential feature of the above method is the eUmination of the con
ditional equations. In Art. 167 the same problem is worked out by eUm
inating the observation equations. The results obtained are of course
identical.
167. Method of Correlatives. The general method of correla
tives is beyond the scope of the present volume. The case here
given is the only one that is likely to be of service to the civil
engineer. In this case the observations are made directly on
each unknown quantity, and the number of observation equations
equals the number of unknown quantities. Let u be the number
of unknown quantities, for which the observation equations may
be written
X = Ml (weight pi);
y = M 2 (weight p^) ,"
t = M„ (weight p J ;
and for which (Art. 164) the conditional equations may be written
aix \a2y...\aJ, = E^
bix + b2y . . . + bj = Eb
mix + may . . . + mj, = E„
(41)
If, as heretofore, x, y, t, etc., be understood to represent the most
probable values of the unknown quantities, and vi, V2, Vu, etc.,
PEOBABLE VALUES OF CONDITIONED QUANTITIES 291
represent the corresponding residuals in the given equations, we
may write
X = Ml + vi (weight pi)
y = M2 + V2 (weight P2)
(42)
t = M^ + Vu (weight pj .
which, substituted in Eq. (41), give the conditional equations
aivi + a2V2 • ■ . + a^Vu = Ea — ^aM
bivi + b2V2 . . . + M« = Eb  26M
miv\ + in2V2 . ■ . + myVy, = E^ — ^mM
(43)
As explained in Art. 164, these conditional equations must be
less in number than the number of unknown quantities. The
values of vi, V2, etc., thus become indeterminate, and an infinite
number of sets of values will satisfy the equations. The values
in any one set (called simultaneous values) are not independent,
however, as they must be such as will satisfy the above equations.
If 2^1, V2, etc., in Eqs. (43) are assumed to vary through all
possible simultaneous values due to any set of values dvi, dv2,
etc., and all possible sets of values dwi, dv2, etc., are taken in turn,
the most probable set of values vi, V2, etc., for the given set of
observations will eventually be reached. The values dvi, dv2,
etc., in any one set, however, can not be independent, as it is
evident that dependent quantities can not be varied indepen
dently. Differentiating Eqs. (43), we have
aidvi + a2dv2 • . . + ciudvu
bidvi + b2dv2
+ b^dvy, =
m\dvi + m2dv2 . . • + mudVy, =
(44)
and these new equations of condition show the relations that must
exist among the quantities dvi, dv2, etc. Since the number of
equations is less than the number of quantities dvi, dv2, etc., it
follows that an infinite number of sets of simultaneous values of
dvi, dv2, etc., is possible. In order to involve Eqs. (44) simul
292
GEODETIC SURVEYING
taneously in an algebraic discussion it is necessary to replace
them by a single equivalent equation, meaning an equation so
formed that the only values which will satisfy it are those which
will individually satisfy the original equations which it replaces.
This is done by writing
ki(aidvi + a2dv2 . . . + audvy)
+ k2{hidvi + h2dv2 . ■  + 6„dt;„)
+ k^imidvi + m2dv2 . ■ ■ + m^dvu)
^ = 0;
(45)
in which ki, k2, etc., are independent constants which may have
any possible values assigned to them at pleasure. Since Eq. (45)
must by agreement remain true for all possible sets of values
ki, k2, etc., its component members must individually remain
equal to zero. But these component members are identical with
the first members of the original conditional equations, so that
no set of values dvi, dv2, etc., can satisfy Eq. (45) unless it can
also satisfy each of Eqs. (44). The values in any such set are
called simultaneous values.
In order to determine the most probable values of vi, V2, etc.,
we must have (Art. 156)
pivi^ + P2V2^ . . . + Pu^u^ = a minimum.
In accordance with the principles of the calculus for the case of
dependent quantities the first derivative of this expression must
equal zero for every possible set of values dvi, dv2, etc. Hence,
by differentiating, and omitting the factor 2, we have
pividvi + p2V2dv2 . . . + PuVudvu = 0,
(46)
in which dvi, dv2, etc., must be simultaneous values. Since these
values are also simultaneous in Eq. (45), we may combine this
equation with Eq. (46) and write
PlVidVi + p2V2dV2 .
+ PuVudVu
ki{aidvi + a2dv2
+ k2(bidvi + b2dv2
. + aJLvu)
. + hydvy)
+ k^imidvi + m2dv2 . . . + m^dv^)
PROBABLE VALUES OF CONDITIONED QUANTITIES 293
whence, by rearranging the terms, we have
[piVi — (AiiOi + k2bi
+ [p2i>2 — {kia2 + ^262
+ kmmi)]dvi
+ kmm2)]dV2
■ '+ [puVu — {kiUu + A;2&« . . . + kmmu)]dvu
= 0.
(47)
Since ki, ^2, etc., are independent and arbitrary constants, it is
evident that this equation can not be true unless its component
members are each equal to zerb, so that
[piVi — (fciai + ^261 . .
etc.,
. from which we have
piVi = kitti + fe&i
P2V2 = kia2 + k2b2
PuK = hau + k2hu
+ k^mi)]dvi = 0;
etc.;
. + fc„m2
+ kj^in^^ .
(48)
as the general equations of condition for the most probable
values of v\, V2, etc.
It is evident that Eqs. (48) can not be solved for vi, V2, etc.,
until definite values have been assigned to ki, fe, etc. In the
general discussion of the problem the values of ki, /c2, etc., have
been entirely arbitrary, since the numerical requirements of
Eqs. (43) vanished in the differentiation. In any particular case,
however, the m conditional Eqs. (43) must be numerically satisfied
in order to satisfy the rigid geometrical conditions of the case,
while the u conditional Eqs. (48) must be satisfied in order to have
the most probable values for vi, V2, etc. There are thus m + u
simultaneous equations to be satisfied. But there are also m \ u
unknown quantities, since the m unknown quatities ki, k2, etc.,
corresponding to the m conditional Eqs. (43), have been added
to the u unknown quantities vi, V2, etc. In any particular case,
therefore, there is but one set of values for the m unknown quan
tities ki, k2, etc., and the u unknown quantitites vi, V2, etc., that
will satisfy the m + u equations consisting of Eqs. (43) and (48).
The auxiliary quantities ki, k2, etc., are called the correlatives
294
GEODETIC SURVEYING
(or correlates) of the corresponding conditional Eqs. (43), and the
quantities vi, V2, etc., are the most probable values of the residual
errors in the observation equations. Substituting in Eqs. (43)
the values of vi, V2, etc., due to Eqs. (48), we have
kill— + feli —
P V
,ani
+ km^— = E,  T.aM
V
JfciS + k2^ ... + k,Jl— =Et  llbM
in which
fciS
am
~P
+ k2^
bm
y ■
.+fc,„S
m2
P
=
E^
p
Pi
P2
+
a^
Pu'
s^ =
aibi
, a2&2
+
ttubu
p
Pi
P2 ■
Pu
etc
)
etc.
HmM
(49)
Attention is called to the fact that the law of the coefficients
in Eqs. (49) is the same (Art. 162) as the law of the coefficients
for normal equations, and this is a check that should never be
neglected. It is evident that fci, ^2, etc., can be found by solving
the simultaneous Eqs. (49). Then, from Eqs. (48), we have
n
V2
ki"^ + k2^
Pi Pi
kl h k2 —
P2 P2
Vu = kl— + ^2"
Pu
Pu
and from Eqs. (42),
+ k,
mi
P2
Pu J
X = Ml + vi
y = M2 + V2
t = M^ + Vu
(50)
(51)
PEOBABLE VALUES OF CONDITIONED QUANTITIES 295
in which x, y, t, etc., are the most probable values of the quantities
whose observed values were Mi, M2, M^, etc.
Example. Given the following data, to find the most probable values of
X, y, and Z''.
fx = 17.82 (weight 2);
Observation equations ] y = 15.11 (weight 4);
[z =29.16 (weight 3).
„ ,.,. , ,. (2x + 5y =112.00;
Conditional equations 1 3^ _j_ y_^ ^ Z^m.
In this case we have
Ea
l^aM =
112.00
111.19
Ei
= 39.00
SbM = 39.41
Ea  SaM =
0.81
E,
S6M =  0.41
Ml = 17.82
Ml = 15.11
Ml = 29.16
oi = 2
Oa = 5
as =0
61 =
62 =
6s =
3 pi =2
1 P2 = 4
 1 Pa = 3
y"^ 33
"p T
Pi
= 1
61 ^3
Pi 2
p 4
£2
P2
_ 5
4
62 1
P2 4
^6^ _ 61
p 12
as
P3
=
6s _ _1
Ps 3
33, , 17,
4 4
17, , 61,
0.81
0.41
giving
ffci = +0.2454.
[fcz = 0.2859.
We thus have
Vi = 0.2454 X 1  0.2859 X I =  0.183;
V2 = 0.2454 X f  0.2859 Xi = + 0.235;
t)3 = 0.2454 X + 0.2859 X i = + 0.t)95;
whence, for the most probable values of x, y, and z, we have
a; = Ml + wi = 17.82  0.183 = 17.637;
y = M2+V2 = 15.11 + 0.235 = 15.345;
s = Ms + fs = 29.16 + 0.095 = 29.255.
As a check on the work of computation we may substitute these values in
the conditional equations, giving
2x+5y = 35.274 + 76.725 = 111.999;
Zx+ 2/  « = 52.911 + 15.345  29.255 = 39.001;
from which it is seen that each equation checks with the corresponding con
ditional equation within 0.001, which is an entirely satisfactory check. The
296 GEODETIC SURVEYING
essential feature of the above method is the elimination of the observation
equations. In Art. 166 the same problem is worked out by eliminating the
conditional equations. The results obtained are of course identical.
168. Most Probable Values of Computed Quantities. By a com
puted quantity is meant a value derived from one or more observed
quantities by means of some geometric or analytic relation.
The most probable values of computed quantities are found from
the most probable values of the observed quantities by employ
ing the same rules that are used with mathematically exact quan
tities. Thus the most probable value of the area of a rectangle
is that which is given by the product of the most probable values
of its base and altitude; the most probable value of the circum
ference of a circle is equal to t: times the most probable value of
its diameter; and so on.
CHAPTER XIII
PROBABLE ERRORS OF OBSERVED AND COMPUTED QUANTITIES
A. Op Observed Quantities
169. General Considerations. The most probable value of
a quantity does not in itself convey any idea of the precision of
the determination, nor of the favorable or unfavorable circum
stances surrounding the individual measurements. Any single
measurement tends to lie closer to the truth the finer the instru
ment and the method used, the greater the skill of the observer,
the better the atmospheric conditions, etc. The accidental errors
of observation tend to be more thoroughly eliminated from the
average value of a series of measurements the greater the number
of measurements which are averaged together. Some criterion
or standard of judgment is therefore necessary as a gage of pre
cision. Since the probability curve for any particular case shows
the facility of error in that case, and thus represents all the sur
rounding circumstances under which the given observations
were taken, it is evident that some suitable function of the proba
bility curve must be adopted as an indication of the precision
of the results obtained. The function which is commonly adopted
as the gage of precision is called the probable error.
170. Fundamental Meaning of the Probable Error. By the
probable error of a quantity is meant an error of such a magnitude
that errors of either greater or lesser numerical value are equally
likely to occur under the same circumstances of observation.
Or, in other words, in any extended series of observations the
probability is that the number of errors numerically greater than
the probable error will equal the number of errors numerically
less than the probable error. The probable error of a single
observation thus becomes the critical value that the numerical
error of any single observation is equally likely to exceed or fall
short of. Similarly the probable error of the arithmetic mean
297
298 GEODETIC SURVEYING
becomes the critical value that the numerical error of any iden
tically obtained arithmetic mean is equally likely to exceed or
fall short of. Thus if the probable error of any angular measure
ment is said to be five seconds, the meaning is that the probability
of the error lying between the limits of minus five seconds and plus
five seconds equals the probability of its lying outside of these
limits. The probable error is always written after a measured
quantity with the plus and minus sign. Thus if an angular
measurement is written
72° 10' 15".8 ± 1".3,
it indicates that 1".3 is the probable error of the given determina
tion. The probable error of a quantity can not be a positive
quantity only, or a negative quantity only, but always requires
both signs. It is important to note that the probable error is an
altogether different thing from the most probable error. Since
errors of decreasing magnitude occur with increasing frequency,
the most probable error in any case is always zero.
171. Graphical Representation of the Probable Error. The
probability that an error will fall between any two given limits
(Art. 147) is equal to the area included between the corresponding
ordinates of the probability curve. The probability that an error
will fall outside of any two given limits must hence be equal to
the sum of the areas outside of these limits. If these two proba
bihties are equal, therefore, each such probability must be
represented by onehalf of the total area. The probable error
thus becomes that error (plus and minus) whose two ordinates
include onehalf the area of the probability curve. Referring
to Fig. 67, the solid ciuve corresponds to a series of observations
taken under a certain set of conditions, and the dotted curve
to a series of observations taken under more favorable conditions.
The ordinates yi, yi, correspond to the probable error n of an
observation of unit weight taken under the conditions pro
ducing the solid probability curve, and include between them
selves onehalf of the area of that curve. The ordinates y', y',
correspond to the probable error r' of an observation of unit
weight taken under the conditions producing the dotted proba
bility curve, and include between themselves onehalf of the
area of the dotted curve. The area for any probability curve
(Art. 150) being always equal to unity, it follows that yi, j/i.
PROBABLE ERROES OF OBSERVED QUANTITIES 299
and y', y', include equal areas. Hence as the center ordinate at
A grows higher and higher with increasing accuracy of observation,
so also must the ordinates yi, yi, draw closer together. It is
thus seen that the probable error n grows smaller and smaller
as the accuracy of the work increases, and therefore furnishes a
satisfactory gage of precision.
Y
Fig. 67.— The Probable Error.
172. General Value of the Probable Error. The area of any
probability curve (Art. 150) equals unity. The area between
any probable error ordinates yi, yi (Art. 171), is equal to half
the area of the corresponding probability curve. But the area
between the ordinates yi, yi (Art. 147), is equal to the probability
that an error will fall between the values x = — ri and x = + ri.
Hence from Eq. (16) we have
1 ^ f '•■
2 VnXrl
e^'^'dx.
(52)
Since (Art. 150) the precision of any set of observations depends
entirely on the value of h, it follows that the probable error ri
must be some fimction of h. The last member of Eq. (52) is not
directly integrable, so that the numerical relation of the quan
tities h and n can only be found by an indirect method of suc
cessive approximation which is beyond the scope of this volume.
As the result of such a discussion we have,
ri =
0.4769363
h
(53)
It is thus seen that for different grades of work the probable error
n varies inversely as the precision factor h.
300 GEODETIC, SURVEYING
By more or less similar processes of reasoning it is also estab
lished that the probable error of any quantity or observation
varies inversely as the square root of its weight. Thus if ri is
the probable error of an observation of unit weight, then for the
probable error rp of any value with the weight p, we have
rp = > (54)
173. Direct Observations of Equal Weight. From Eq. (20)
we have
^ ~\ 25:*>2 •
Substituting this value of h in Eq. (53) and reducing, we have
n = 0.6745 ^^^zTi' ^^'"^
in which r\ is the probable error of a single observation in the
case of direct observations of equal weight on a single unknown
quantity, and n is the number of observations.
Since in this case (Art. 134) the weight of the arithmetic
mean is equal to the number of observations, we have (Art. 172),
Ta 77= =0.6745. , '^■,. , . . . (56)
Vn y^nin — 1)
in which r^ is the probable error of the arithmetic mean in the
case of direct observations of equal weight on a single unknown
quantity, and n is the number of observations.
Example. Direct observations on an angle A :
Observed values v
29° 21' 59".l  2".l
29 22 06 .4 +5 .2
29 21 58 .1  3 .1
3 )88 06 03 .6
z = 29 22 01 .2
The probable error of a single observation is therefore
n = 0.6745"^^ = 0.6745 V^= ± 3".06;
»2
4.41
27.04
9.61
Sw^
= 41.06
n
= 3
PROBABLE EREOES OF OBSERVED QUANTITIES 301
and of the arithmetic mean,
n 3.06
whence we have
= ± 1".76;
Most probable value of A = 29° 22' 01".2 ± 1".76.
174. Direct Observations of Unequal Weight. From Eq. (2l)
we have
Substituting this value of h in Eq. (53) and reducing, we have
n = 0.6745 J— ^, (57)
in which ri is the probable error of an observation of unit weight
in the case of direct observations of unequal weight on a single
unknown quantity, and n is the number of observations. The
value of n thus becomes purely a standard of reference, and it is
entirely immaterial whether or not any one of the observations
has been assigned a unit weight. Having found the value of
ri we have, from Eq. (54),
_ '"1
in which rp is the probable error of any observation whose weight
is p.
Since in the case of weighted observations (Art. 134) the weight
of t^e weighted arithmetic mean is equal to the sum of the indi
vidual weights, we have (Art. 172),
^^=vk^'''^'4^^^)' ■ ■ ■ ^''^
in which rpa is the probable error of the weighted arithmetic mean
in the case of direct observations of unequal weight on a single
imknown quantity.
302 GEODETIC SURVEYING
Example. Direct baseline measurements of miequal weight:
Observed values p pM v v^ pifl
4863.241ft. 2 9726.482 0.020 0.000400 0.000800
4863.182 ft. 1 4863.182  0.039 0.001521 0.001521
Sp = 3 )14589.664 Spu' = 0.002321
2=4863.221 ft. n = 2.
The probable error pf an observation of miit weight is therefore
0.6745 J^^= 0.6745x1 """^'^^^ ±0.032 ft.;
■ 'n — 1 ' ^ 1
of an observation of the weight 2,
r, 0.032
rj = 4= = — = = ± 0.023 ft.;
Vp V2
and of the weighted arithmetic mean,
r, 0.032 „„ ,
rpa = ^=i= y=^ ± 0.019 ft.;
whence we have
Most probable value = 4861.221 ± 0.019 ft.
175. Indirect Observations on Independent Quantities. From
Eq. (22) we have
»=j;
 Q
Substituting this value of h in Eq. (53) and reducing, we have
n =0.6745^^, (59)
in which ri is the probable error of an observation of unit weight
in the case of indirect observations on independent quantities
(that is with no conditional equations), n is the number of observa
tion equations, and q is the number of unknown quantities.
Having found the value of n, we have, from Art. 172,
n n n .
rv = —i=, r^ = —.= , Ty = —=, etc.,
vp Vp^ Vpy
in which rp is the probable error of a;ny observation whose weight
is p, and r^ is the probable error of any unknown, x, in terms of
its weight p^, and so on.
PEOBABLE EERORS OF OBSERVED QUANTITIES 303
The weights px, Pv, etc., of the unknown quantities are found
from the normal equations by means of the following
Rule: In solving the normal equations preserve the absolute
terms in literal form; then the weight of any unknown quantity is
contained in the expression for that quantity, and is the reciprocal
of the coefficient of the absolute term which belonged to the normal
equation for that unknown quanti y.
In applying the above rule no change whatever is to be made
in the original form of any normal equation until the absolute
term has been replaced by a literal term. If the normal equations
are correctly solved the coefficients in the literal expressions for
the unknown quantities will follow the same law (Art. 162) as
the coefficients of normal equations, and this check must never
be neglected.
Example. Given the foUowing observation equations to determine the
most probable values and the probable errors of the unknown quantities: .
z + y = 10.90 (weights);
2x  y = 1.61 (weight 1);
X + 3y = 24.49 (weight 2).
Forming the normal equations, we have
9a; + 7y = 84.90 = N^ = normal equation in x;
7x + 22y = 178.03 = JVj, = normal equation in y;
whence
X = tANx  TTsNy = 4.172, nearly;
y ThNx + iTwNy = 6.765, nearly;
and, by the rule, '
Weight of a; = W = 6.773, nearly = p^;
" 2/ = i^ = 16.556 " =py.
Substituting in the original equations the values obtained for x and y, there
results
x+ y = 10.937;
2x  y = 1.579;
x + 3y = 24.467;
whence, for the residuals, we have,
Vi = 10.90  10.937 =  0.037 (weight 3);
V2 = 1.61  1.579 = + 0.031 (weight 1);
V, = 24.49  24.467 = + 0.023 (weight 2).
We therefore have for the probable error of aa observation of unit weight.
0.6745 J^^^ = 0.6745a/^555^ = ± 0.053;
'71 — a '3—2
304 GEODETIC SURVEYING
for the probable error of x,
■Ti 0053
r, = — F= = , = ± 0.020;
Vp^ V6773
and for the probable error of y,
tx 0.053 „„,„
rj, = — ^ = , = ± 0.013;
Vpy V16.556
whence we write
X = 4.172 ± 0.020 and y = 6.765 ± 0.013.
176. Indirect Observations Involving Conditional Equations.
From Eq. (23) we have
g + c
Substituting this value of In. in Eq. (53) and reducing, we have
'^/^
ri = 0.6745^/^^^, .... (60)
in which r\ is the probable error of an observation of unit weight
in the case of indirect observations involving conditional equa
tions, n is the number of observation equations, q is the niunber
of unknown quantities, and c is the number of conditional equa
tions. Having found the value of r\, we have, from Art. 172,
_ ri _ ri _ ri
Tf — ,_, r^ — , — , Ty =, etc.,
in which, as in the previous article, r^ is the probable error of any
observation whose weight is p, and r^ is the probable error of any
unknown, x, in terms of its weight 'p^, and so on.
In order to find the value of the weights p^,, p^, etc., the con
ditional equations are first eliminated (Art. 166), and the normal
equations due to the resulting observation equations are then
treated by the rule of the preceding article. By repeating the
process with different sets of unknowns eliminated, the weight
of each unknown will eventually be determined.
177. Other Measures of Precision. The measures of precision
thus far introduced are the precision factor h,, and the probable
error r. Two other measures of precision are sometimes used,
PROBABLE EEEOES OF OBSERVED QUANTITIES 305
and are of great theoretic value. These are known as the mean
error, &nd.\hQ mean absolute error. _  ^
By the'mean error is meant the square root of the sum of the^
squares^ of the true errors.  —
By the mean absolute error (often called the vwan of the errors)
is meant the arithmetic mean of the absolute values (numerical
values) of the true errors. _
Referring to Fig.' 68, the precision factor h is equal to Vtt
times the central ordinate AY. Considering either half of the
Poiat of Inflection
Point of Inflection
Fig. 68. — Measures of Precision.
curve alone, the ordinate for the probable error r bisects the
included area, the ordinate for the mean absolute error i) passes
through the center of gravity, and the ordinate for the mean
error e passes through the center of gyration about the axis A Y.
The ordinate for e also passes through the point of inflection
of the curve.
The measure of precision most commonly used in practice is
the probable error r, but as the different measures bear fixed
relations to each other a knowledge of any one of them determines
the value of all the others, as shown in the following summary:
Precision factor = h.
Probable error = r =
0.4769363
Mean absolute error = rj
Mean error
hVn
1
= 1.1829 r.
= 1.4826 r.
306 GEODETIC SURVEYING
B. Of Computed Quantities
178. Typical Cases. When the probable error is known
for each of the quantities from which a computed quantity is
derived, the probable error of the computed quantity may also
be determined. Any problem which may arise will come under
one or more of the five following cases :
1. The computed quantity is the sum or difference of an observed
quantity and a constant.
2. The computed quantity is obtained from an observed quantity
by the use of a constant factor.
3. The computed quantity is any function of a single observed
quantity.
4. The computed quantity is the algebraic sum of two or more
independently observed quantities.
5. The computed quantity is any function of two or more inde
pendently observed quantities.
The fifth case is general, and embraces all the other cases.
The first four cases, however, are of such frequent occurrence that
special rules are developed for them . Any combination of the rules
is therefore admissible that does not violate their fundamental
conditions, since the first four rules are only special cases of the
fifth rule.
179. The Computed Quantity is the Sum or Difference of an
Observed Quantity and a Constant.
Let u and r„ = the computed quantity and its probable error;
X and r^ = the observed quantity and its probable error;
a = a constant;
then
and
u — ± X ± a;
ru = r^ (61)
It is evidently immaterial whether x is directly observed or
is the result of computation on one or more observed quantities.
The only essential condition is satisfied if r^ is the probable error
of X. If a; is a computed quantity the probable error r^ may be
derived by any one of the present rules.
PROBABLE ERRORS OF COMPUTED QUANTITIES 307
Example. Referring to Fig. 69, the most probable value of the angle x is
X = 30° 45' 17".22 ± 1".63.
What is the most probable value of its supplement y, and the probable error
of this value 7
From the conditions of the problem we
have
y = 180°  x;
whence
?•„ = r^ = ± 1".63, F:g. 69.
and
y = 149° 14' 42".78 ± 1".63.
180 The Computed Quantity is Obtained from an Observed
Quantity by the Use of a Constant Factor.
Let u and r„ = the computed quantity and its probable error;
X and Tx = the observed quantity and its probable error;
a = a constant;
then
and
u = ax
Tu = ar^ (62)
Evidently, as in the previous case, x may be any function of one
or more observed quantities, provided that r^is its correct probable
error. The rule of this article is only true when the constant a
represents a strictly mathematical relation, such as the relation
between the diameter and the circumference of a circle. Staking
out 100 feet by marking off successively this number of single
feet is not such a case, as the total space staked out is not neces
sarily exactly 100 times any one of the single spaces as actually
marked off. In all probability some of the feet will be too long
and others will be too short, so that (owing to this compensating
effect) the total error will be very much less than 100 times any
single error, and the probable error must be found by Art. 182.
In the case of the circle, however, the circumference is of neces
sity exactly equal in every case to n times the diameter.
308 GEODETIC SURVEYING
Example. The radius of a circle, as measured, equals 271.16 ± 0.04 ft.
What is the most probable value of the circumference, and the probable
error of this value?
Circumference = 271.16 X 2x = 1703.75 ft.;
r^= r^X 2x = ± 0.04 X 2x = ± 0.25 ft.;
whence we write
Circumference = 1703.75 ± 0.25 ft.
181. The Computed Quantity is any Function of a Single
Observed Quantity.
Let u and r„ =the computed quantity and its probable error;
X and r^ = the observed quantity and its probable error;
then
u = (j>(x);
and
'='^t (^^)
Evidently, as in the two previous cases, x may be any function
of one or more observed quantities, provided that r^ is its correct
probable error.
Example. The radius 5 of a circle equals 42.27 ± 0.02 ft. What is
the most probable value and the probable error of the area?
M = «e' = (42.27)^ X X = 5613.26;
du = 2xxdx, — = 2xa;,
dx
■ru = 'rx~ = rx{2%x) = ± 0.02 X 2x X 42.27 = ± 5.31;
dx
whence we write
Area = 5613.26 ± 5.31 sq.ft.
182. The Computed Quantity is the Algebraic Sum of Two or
More Independently Observed Quantities.
Let u and r„ = the computed quantity and its probable
error;
X, y, etc. = the independently observed quantities;
Tx, Ty, etc. = the probable errors of x, y, etc. ; J
then
w = ± a; ± 2/ ± etc.;
and
/•uVrJ+V + etcT ...... (64)
PROBABLE EEEOES OF COMPUTED QUANTITIES 309
The observed quantities x, y, z, etc., may each be a different
function of one or more observed quantities, but the absolute
independence of x, y, z, etc., must be maintained. In other
words, X must be independent of any observed quantity involved
in y, z, etc.; y independent of any observed quantity involved
in X, z, etc. ; and so on. Thus, for instance, we can not regard
2x as equal to a; + a;, and substitute in the above formula, since
X and X in the quantity 2x are not independent quantities.
Attention is also called to the fact that the signs under the
radical are always positive, whether the computed quantity is
the result of addition or subtraction or both combined.
Example 1. Referring to Fig. 70, given
X = 70° 13' 27".60 ± 2".16;
2/ = 40 67 19 .32 ± 1 .07;
to find the most probable value and the probable error of z.
In this case
2 = x + y = 111° 10' 46".92;
whence we write
r„ = V (2.16)^ +(1.07)' = ± 2".41;
2 = 111° 10' 46".92 ± 2".41.
^
<^1^ V^
Fig. 70.
Fig. 71.
Example 2. Referring to Fig. 71, given
X = 70° 13' 27".60 ± 2".16;
3/ = 40 57 19 .32 db 1 .07;
to find the most probable value and the probable error of z.
In this case
2 = a;  2/ = 29° 16' 08".28;
r„ = V'(2.16)' + (1.07)' = ± 2".41;
whence we write
z = 29° 16' 08".28 ± 2".41.
310 GEODETIC SURVEYING
183. The Computed Quantity is any Function of Two or More
Independently Observed Quantities.
Let u and r„ = the computed quantity and its probable error;
X, y, etc. = the independently observed quantities;
r^, Ty, etc. = the probable errors of x, y, etc.;
then
u = ^{x, y, etc.);
and
^" = V(^^^)'+('"«S)' + ^*^ • • • ^^^^
All the remarks under the previous case apply with equal force
to the present case.
Example 1. The measured values for the two sides of a rectangle are
X = 55.28 ± 0.03 ft.
y = 85.72 ± 0.05 ft.
What is the most probable value of the area and its probable error?
u = xy = 55.28 X 85.72 = 4738.60;
du _ du _
dx ' dy '
Tu = ^{rxyY + (ryxY
= V(0.03 X 85.72)^ + (0.05 X 55.28)^ = ± 3.78;
whence we write
Area = 4738.60 ± 3.78 sq.ft. m
Example 2. Referring to the rightangled
triangle in Fig. 72, given
x = 38.17 ± 0.06 ft.; Fig. 72.
y = 19.16 ± 0.04 ft.;
to find the most probable value of the hypothenuse u and its probable error.
M = Va;^ + 2/2 = V(38.17)2 + (19.16)' = 42.71;
du _ X du _ y
dx Va:' + 2/2 'dy V x^ + y^'
{rxxy + (r„yy
x' + y'
r = K j'^. y + ( 'yy V = /'
/ (38.17 X 0.05)2 + (19.16 X 0.04)' ^
V (38.17)2 + (19.16)2 ±0.05;
PROBABLE ERRORS QF COMPUTED QUANTITIES 311
whence we write
Hypothenuse = 42.71 ± 0.05 ft.
Example 3. Referring to Pig. 73, in which the horizontal distance x
and the vertical angle = 12° 17' ± 1'; Fig. 73.
to find the most probable value of the elevation u and its probable error
w = a; tan (ji = 106.49;
du X
du
dx
tan 0,
d
Example 2. Referring to Fig. 77, the following observations are to be
adjusted ;
X = 40° 16' 23".7 (weight 2);
2/ = 46 36 48 .5 (weights);
a; + 2/ = 86 53 08 .0 (weight 4).
40° 16' 23".7
46 36 48 .5
86 53 12 .2
86 53 08 .0
d = 04 .2
In accordance with the above principles this discrepancy is to be distributed
numerically as
1 . i . 1.
2 '3 "4'
which, cleared of fractions, equals
6:4:3;
giving as the most probable corrections .
 4.2 X fV =  1".94
 4.2 X T^ =  1".29
+ 4.2 X A = + 0".97
and therefore as the most probable values
X = 40° 16' 21".76;
2/ = 46 36 47 21;
a; + 2/ = 86 53 08 .97.
191. The General Case. The cases given in Arts. 188, 189, and
190, are the only ones in which it is desirable to establish special
rules. Any case of station adjustment may be solved by writ
ing out the observation and conditional equations and then apply
ing the principles developed in Chapters XI and XII.
320
GEODETIC SURVEYING
Example 1. Referring to Fig. 78, find the most probable values of the
angles x, y, and z, from the foUowing observations :
s + 2/ = 53
X + y + z =86
X = 25° 17' 10".2 (weight 1)
2/ = 28 22 16 .4 (weight 2)
z = 32 40 28 .5 (weight 2)
39 23 .1 (weight 2)
19 57 .8 (weight 1).
Letting v,, vi, v,, be the most probable corrections for x, y, and z, we may
write (Art. 163) the reduced observation equations
Vi = 0".0 (weight 1)
V2 = .0 (weight 2)
v, = .0 (weight 2)
Vi + Vi = — 3 .5 (weight 2)
V, + vi + V, = + 2 .7 (weight 1)
Fig. 78.
Fig. 79.
from which we have the normal equations
» vi + 3w2 + 1)3 =  4.3
3wi + 51)2 + V3 = — 4.3
vi+ V2+ Svs = 2.7
whose solution gives
Vi =  1".04, V2=  0".52, Ds
+ 1".42.
The most probable values of the given angles are therefore
X = 25° 17' 09".16;
y = 28 22 15 .88;
« = 32 40 29 .92.
Example 2. Referring to Fig. 79, find the most probable values of the
angles x, y, and z, from the following observations:
X = 14° 11' 17".l (weight 1);
y = 19 07 21 .3 (weight 2);
X + y = 33 18 43 .4 (weight 1);
z = 326 41 18 .2 (weight 2);
y + z = 345 48 39 .2 (weight 3).
APPLICATION TO ANGULAR MEASUREMENTS 321
As the angles x, y, and z close the horizon they must satisfy the conditional
equation
x + y + z = 360°.
Avoiding this conditional equation by subtracting all angles containing
z from 360°, we have
X = 14° 11' 17".l (weight 1);
2/ = 19 07 21 .3 (weight 2);
X + 2/ = 33 18 43 .4 (weight 1);
X + 2/ = 33 18 41 .8 (weight 2);
I = 14 11 20 .8 (weight 3);
in which x and y may be regarded as independent quantities.
Letting v^ and Vi be the most probable corrections for x and y, and
writing the reduced observation equations in accordance with Art. 163,
we have
wi = 0".0 (weight 1);
tij = .0 (weight 2);
vi + Vi = 5 .0 (weight 1);
vi + vi = Z A (weight 2);
vi = 3 .7 (weights);
from which we have the normal equations
7j)i + 3f2 = 22.9;
3»i + 5v2 = 11.8;
whose solution gives
vi= \ 3".04, wj = + 0".53.
The most probable values of x and y are therefore
X = 14° 11' 20".14;
y = 19 07 21 .83;
and hence the most probable value for z must be
z = 326° 41' 18".03,
in order to make the sum total of 360°
Figure Adjustment
192. General Considerations. All cases of figure adjust
ment necessarily imply one or more conditional equations. In
the determination of the most probable values of the several
angles these equations may be avoided (Art. 165), eliminated
(Art. 166), or involved in the computation (Art. 167), as found
most convenient. The angles in a triangulation system are in
general measured under similar conditions, so that in making the
adjustment it is customary to give to each angle a weight equal to
the number of observations (or the sum of the weights in the case
of weighted observations) on which it depends. Angles are sel
dom measured a sufficient number of times to make it justifiable
to weight them inversely as the squares of their probable errors,
322 GEODETIC SUEVEYING
as would be required by the last paragraph of Art. 172. In work of
moderate extent any required station adjustment may be made
prior to the figure adjustment, but in very important work it may
be desirable to make both adjustments in one operation. Except
in very important work, the triangles, quadrilaterals, or other
figures in a system may be adjusted independently. In work of
the highest importance the whole system would be adjusted in
one operation. The following cases of figure adjustment show
the general principles involved, assimiing that the reduction for
spherical excess (Arts. 66, 57, 58) has already been made.
193. Triangle Adjustment with Angles of Equal "Weight.
Referring to Fig. 80,
Fig. 80.
Let X, y, z = the unknown angles;
a,b,c = the measured values;
d = (a + b { c) — 180° = the discrepancy to be
adjusted.
Avoiding the conditional equation (Art. 163) for the sum of the
three angles by writing the observation equations in terms of
X and y as independent quantities, we have
X = a;
y = b;
x + y = 180°  c.
Substituting for the most probable values
X = a \ vi;
y = b + V2;
we have
vi =0;
V2 = 0;
i^i + «'2 = 180°  (a + b + c) =  d;
APPLICATION TO ANGULAE MEASUEEMENTS 323
giving the normal equations,
2^1 + V2 = — d;
vi + 2^2 = — d;
whence by subtraction,
v\ — V2 = 0, or vi = Vi
la, a similar manner it may be shown that vi or v^, is equal to
vz, or in general,
Vi = V2 = V3.
Bat evidently,
vi {■ V2 [ Vz = — d;
whence,
i;i = ■y2 = ■y.s =  (68)
Equation (68) shows that when the measured angles of a tri
angle are considered of equal weight, the most probable values of
these angles are found by adjusting each angle equally for onethird
of the discrepancy.
Example. The measured values (of equal weight) for the three angles
of a triangle are 92° 33' 15".4, 48° 11' 29".6, and 39° 15' 12".3. What are
the most probable values?
Measured Values Most Probable Values
92° 33' 15".4 92° 33' 16".3
48 11 29 .6 48 11 30 .5
39 15 12 .3 39 15 13 .2
179° 59' 67".3 180° 00' 00" .0
180 00 00 .0
3)  02".7
 0".9 '
194. Triangle Adjustment with Angles of Unequal Weight.
Referring to Fig. 80,
Let X, y, z = the unknown angles;
a,b,c== the measured values;
Pi) P2, Pz — the respective weights;
d = (a + 6 f c) — 180° = the discrepancy to be
adjusted.
Avoiding the conditional equation as before by making x and y
the independent quantities, we have
a; = a (weight pi);
y = b (weight ^2);
X + y = 180° — c (weight pz).
324
GEODETIC SUEVEYING
Substituting, as before,
we have
= (weight pi);
j;2 = (weight p2);
VI+V2 = 180° {a + b + c) = d (weight ps);
giving the normal equations
Pivi + psivi + V2) = — Pad;
P2V2 + psivi + V2) = — pid;
whence, by subtraction,
piVi — P2V2 = 0, or piVi = P2V2
In a similar manner it may be shown that pivx or P2V2 is equal to
P3V3. Hence, in any case,
vi + V2 + 1)3= — d]
PlVi = P2V2= P3V3.
(69)
Eqs. (69) show that when the measured angles of a triangle are
considered of unequal weight, the most probable values of these
angles are found by distributing the discrepancy inversely as the
corresponding weights.
Example. The measured values for the three angles of a triangle axe
97° 49' 56" .8 (weight 2), 38° 06' 05".0 (weight 1), and 44° 04' 01".l (weight 3).
What are the most probable values?
97° 49' 56".8
38 06 05 .0
44 04 01 .1
180° 00' 02".9
180 00 00 .0
i: = S,e.;
d = + 02".9
3+6 + 2 = 11;
+ 02.9 X A = + 00".79, + 02.9 X 1^ = + 01".58,
+ 02.9 X A = + 00".53.
The most probable values are therefore
97° 49' 56".01
38 06 03 .42
44 04 00 .57
180° 00' 00".00
APPLICATION TO ANGULAB MEASUEEMENTS 325
195. Two Connected Triangles. A simple case of figure
adjustment is illustrated in Fig. 81. Two triangles are here
connected by the common side AB, and the eight indicated
angles are measured. It is evident from the figure that four
independent conditional equations must be satisfied by the
adjusted values of the angles, for the summation angles at A and B
must agree with their component angles, and the angles in each
of the two triangles must add up to 180°. The problem may be
worked out by the methods of Arts. 165, 166, or 167. The fol
A
Fig. 81. — ^Two Connected Triangles.
lowing example is worked out by the algebraic elimination of the
conditional equations (Art. 166) in order to illustrate this method.
fee.
Exampk. Referring to Fig. 81, given the following observed values of
equal weight, to find the most probable values of the measured angles:
observed Values of Angles
Ai = 65° 25' 18".l; A = 141°
A2 = 75 43 45 .1; B = 100
Bs = 47 26 11 .9; C = 67
Bi = 53 19 51 .8; D = 50
09'
46
08
56
02'
06
28
25
•2;
.6;
•4;
.2.
t the four conditional equations, we have
A = Ai+ Ai\
B = B,+ Bi,
C+Ai+B, = 180°;
D +A2 + Bi = 180°.
In accordance with Art. 166, any four of the imknowns which may be
considered as independent may be found from these equations in terms
of the remaining unknowns. It is evident from an inspection of either the
figure or the conditional equations that A, B, C, and D may be thus con
sidered as independent. These four are selected in preference to any other
326
GEODETIC SURVEYING
four because they are so easily found from the given conditional equations.
Solving for these quantities, we have
A = Ai+ Ai,
B = Bs \ Bt;
C = 180°  (A1 + B3);
D = 180°  (A2 + B4).
Substituting in the observation equations and reducing, we have
Ai = 65°
25'
18".l;
Ai + ^2 = 141°
09'
02".2;
Ai = 75
43
45 .1;
B3 + B, = 100
46
06 .6;
B3 =47
26
11 .9;
Ai + B3 = 112
51
31 .6;
Bi = 53
19
51 .8;
A2 + B4 = 129
03
34 .8.
Letting Vi, vi, Vs, Vi, be the most probable corrections for Ai, A2, B3, B4,
respectively, we may write the reduced observation equations (Art. 163)
as follows:
Vl = 0".0
!)2 = .0
Us = .0
U4 = .0
fl + W2 =  1".0
fa + ^4 = + 2 .9
vi + va = +\ .6
Vi \ Vl = — 2 .1.
In a simple case like this the reduced observation equations would usually
be written directly from the figure instead of going through the above alge
braic work. Having decided on the proper independent quantities, these
equations are simply written so as to represent the apparent discrepancy
in each observation, always subtracting the independent quantities from
the values they are compared with. Forming the normal equations, we have
whose solution gives
Using these corrections to find Ai, A2, B,, and B4, and then the conditional
equations to find A, B, C, and D, we have for the most probable values
Svi + Da + W3
Vl + 3Vi + 1)4 =
Vl + 3W3 + 2)4 =
2)2 + 1)3 + 32)4 =
= + 0".6;
= 3 .1;
= +4 .5;
 + .8;
Vl = + 0".10, 2)3 =
!)2 = — 1 .13, Vl =
 + 1".41,
= + .17.
Ai = 65° 25' 18".20
A2 = 75. 43 43 .97
B3 = 47 26 13 .31
Bi = 53 19 51 .97
A =141° 09' 02".17;
B = 100 46 05 .28;
C = 67 08 28 .49;
D = 50 56 24 .06.
196. Quadrilateral Adjustment. The best method to use in
adjusting a geodetic quadrilateral, Fig. 82, is the method of
correlatives, Art. 167. In accordance with Art. 58 the adjusted
angles must satisfy the following three angle equations:
a + b + c + d+e+r+g + h=360° ]
a+b=e+f ' .... (70)
c+d=g+h ^ J
APPLICATION TO ANGULAR MEASUREMENTS 327
and also the following side equation :
sin a sin c sin e sin 5f_
sin 6 sin d sin /sin A '
which naay be written in the logarithmic form
S log sin(a, c, e,g)— S log sin(6, d,j, h)
(71)
(72)
Fig. 82. — ^The Geodetic Quadrilateral.
Letting Ma, Mj,, etc., represent the measured values of the
angles a, b, etc., and h, h, h, h, represent the discrepancies in
these equations due to the errors in the measured angles, we have
S(M„toMA)360° = Zi
S log sin (ikf„, M„ Me, Mg)  2 log sin {M^, Ma, Mf, M^) = h 
The corrections Va, v^, etc., to be added algebraically to the
measured values Ma, Mj,, etc., must reduce these equations to
zero in order that the conditional equations (70) and (71) may be
satisfied. Therefore we must have
Va\ Vi+ Vo + Vd+ Ve+ V/+ Vg+ Vh= li '
Va+ Vb  Ve Vf = h
Vc + Vd — T^g— ^h= h
daVa—diVb + dcVc—ddVd + deVedfV/+dgVgdhVh= —h ■
in which Va, Vf,, etc., are to be expressed in seconds, and in which
da, db, etc., are the tabular differences for one second for the
(73
328
GEODETIC SUEVEYING
log sin Ma, log sin Mi,, etc. If any angle is greater than 90° it
is evident that the corresponding tabular difference must be
considered negative, since the sine will then decrease as the angle
increases in value. The conditional Eqs. (73) being in the form
of Eqs. (43), the most probable values of Va, Vt, etc., may now
be found by the method of correlatives (Art. 167), by means of
Eqs. (49) and (50). Rewriting these equations with the symbols
used in the present article, and remembering that there are four
conditional equations and hence four correlatives required, we
have in the general case, from Eqs. (49) and (73),
, ^a^ J y^ab „ac „ad
P P P P
■h
k^^"b b^ be bd^_
P P P P
kii:+k2 2 + ksJl + ki^ = h
p p P p
kX + k2i:^^ + ksii'^ + k,i:'^=^
p p
and from Eqs. (50) and (73),
Va
Vb
h
= «! h k2—
Pa Pa
Pa
= ki \ A2 —
Pb Pb
kj^
Pb
Pc
+ Ala + ki^
Pc Pc
Pd
+ k,^  k,^
Pd yd
7 ^ 7 1
= ki k2—
Pe Pe
+%*
7 1 7 1
'4
Pf
Pa
 kS + kiii
Po Pg
Vh = ki~
Ph
k„
Ph J
(74)
(75)
APPLICATION TO ANGULAR MEASUREMENTS 329
(76)
in which p„ represents the weight of lf„, p^ the weight of M^, and
so on.
In the case of equal weights we have, from Eqs. (73) and (74),
8h + [ida + d,+de + dg)  {di, + dd + df+dh)]k4: = h
4k2+ {da db de + d/)ki = — h
4A;3 + {dcdddg+dh)ki = h
[(da + dc+de + dn)  {di,+dd + df+dk)]h
+ {dadbd^ + df)k2+{dcdadg+dh)k3+'2d^k4= Z4 .
and from Eqs. (75),
Va = kl + k'j + daki
1% = ki + k2 — diki
Vc = ki + ks + dcki
'Vd = ki + ks — ddki
Ve = ki — /C2 + deki
Vf = ki — k^ — dfki
Vo = ki — ks + dgki
Vh = ki  ks  dhki
Having found the values of Va, Vi, etc., we have in any case for
the most probable values of the angles a, b, etc.,
a = Ma + Va, e = Me + v^;
b=Mb + vi,; f=M,+Vr;
c = Mc + Vc, g = Mg + Vg;
d= Md + Va; h= Mu + Vh.
(77)
\
(78)
197. Other Cases of Figure Adjustment. There is evidently
no limit to the number of cases of figure adjustment that may be
made the subject of consideration, but few of them are likely to
be of interest to the civil engineer. Any case that may arise may
be adjusted by the method of correlatives (Art. 167), similarly to
the quadrilateral adjustment (Art. 196), provided the observa
tion equations and conditional equations are properly expressed.
In any case the conditional equations must cover all the geo
metrical conditions which must be satisfied, and at the same time
must be absolutely independent of each other. The number of
330
GEODETIC SURVEYING
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If a base line is measured
from end to end a number of times in the same manner, but under
333
334
GEODETIC SURVEYING
such conditions that the different determinations of its length
must be regarded as of unequal weight, then (Art. 157) the weighted
arithmetic mean of the several results is the most probable value
of its length. The probable error of a single measurement of
unit weight (Art. 174) is given by the formula
ri = 0.6745
(81)
the probable error of any measurement of the weight p (Art. 174)
by the formula
rp=—^ = 0.6745
Vp
4.
lipv^
p{n — 1)
(82)
and the probable error of the weighted arithmetic mean (Art. 174)
by the formula
rj,„ = ^ = 0.6745
4
Tipv^
Sp. {n — ly
(83)
Example. Direct baseHne measurements of unequal weight:
Observed Values
7829.614 ft.
7829.657 ft.
7829.668 ft.
7829.628 ft.
pU
7829.614
15659.314
7829.668
23488,884
 0.026
+ 0.017
+ 0.028
0.012
0.000676
0.000289
0.000784
0.000144
0.000676
0.000578
0.000784
0.000432
2p
7 )54807.480
^z= 7829.640 ft.
n = 0.6745,
0.0194
Upv^ = 0.002470
n = 4
^/'
002470
± 0.0194 ft.
n =
rpa =
V2
0.0194
± 0.0137 ft.
^ = ± 0.0112 ft.
V3
0.0194
VY
± 0.0073 ft.
Most probable value = 7829.640 ± 0.0073 ft.
200. Duplicate Lines. In work of ordinary importance or
moderate extent it is sufficient to measure a base line twice and
average the results for the adopted length. When the same line
APPLICATION TO BASELINE WORK 335
is measured twice with equal care it is called a duplicate line. The
rules of Art. 198 necessarily include duplicate lines, but this
case is of such frequent occurrence that special rules are found
convenient for the probable errors. Letting d represent the dis
crepancy between the two measurements, and remembering that
the arithmetic mean is the most probable value, we have
d , d
''i = + 2 ^^^ ''s =  2 •
Substituting these values in Eq. (79) and replacing n with Vi for
the case of duplicate lines, we have for the probable error of a
single measurement of the length I,
n = 0.4769Vd2 = ± 0.4769d. . . . (84)
Substituting the same values in Eq. (80), we have for the probable
error of the arithmetic mean,
r„ = ± 0.3348 d; (85)
Ta (approximately) = ± ^d (86)
whence
Example. Measurement of a duplicate base line:
Observed Values
4998.693 ft. 0.4769 X 0.034 = 0.0162.
4998.659 ft. 0.3348 X 0.034 = 0.0114.
d = 0.034 ft.
r, = ± 0.0162 ft. ra= ± 0.0114 ft.
Most probable value = 4998.676 ± 0.0114 ft.
201. Sectional Lines. A base line may be divided up into
two or more sections, and each section measured a number of
times as a separate line. Each section, on account of its several
measurements, will thus have a most probable length and a prob
able error independent of any other section of the line. If
lijh, ■ • • ^K)be the most probable lengths of the several sections,
then (Art. 168) the most probable length L for the whole hne, is
L = h + h ... + ln = ^l (87)
And if ri, /•2, . . . »•„, be the probable errors of the several values
li, h, etc., then (Art. 182) the probable error tl for the whole
line, is
ri = Vri2 + rgZ . , . + r„2 = VSr^. . . . (88)
336 GEODETIC SURVEYING
Example. Sectional baseline measurement. Given
k = 3816.172 ± 0.022 ft.
k = 4122.804 ± 0.019 ft.
h = 3641.763 ± 0.017 ft.
L = 3816.172 + 4122.804 + 3641.763 = 11580.739 ft.
r^ = \/(0.022)' + (0.019)^ + (0.017)^ = ± 0.034 ft.
Most probable value L = 11580.739 ± 0.034 ft.
202. General Law of the Probable Errors. In measuring a
base line bar by bar or tapelength by tapelength, the case is
essentially one of sectional measurement (Art. 201), in which
each section is measured a single time, and in which each full
section is of the same measured bar or tapelength. If the con
ditions remain unchanged throughout the measurement, therefore,
the probable error will be the same for each full section. As
explained in Art. 180, however, this is not a case of computed
values depending on a constant factor, so that the probable error
of the whole line will not follow the law of that article.
Let L = the total length for a line of full sections;
tl = theprobableerror of this line;
t = the length of the measuring instrument;
Tt = the probable error for each length measured;
n = the number of lengths measured;
then (Art. 201)
r^ = \/Sr2 = Vnr?.
But evidently
L
" = T =
whence
 = Vr^^^^^
(89)
Eq. (89) is derived on the assumption that only full bar or tape
lengths are used. The fractional lengths that occur at the ends
of a base (or elsewhere) form such a small proportion of the total
length, however, that no appreciable error can arise by assuming
Eq. (89) as generally true. A consideration of the various
methods and instruments used in measuring base lines also shows
APPLICATION TO BASELINE WORK 337
that there is nothing in any case which can materially modify
the truth of this equation. We may therefore write as a
General Law : Under the same conditions of measurement the
probable error of a base line varies directly as the square root of its
length.
From the manner in which this law has been derived it is
evident that it is theoretically true whether the length assigned
to;a base line is the result of a single measurement, or. the average
of a number of measurements, so long as the lines being compared
have all been measured in the same way. In cases where the
given lines have been measured more than once, so that each
line has its own direct probable error, we can not expect an exact
agreement with the law. But this /elation of the probable
errors is more likely than any other that can be assigned, and
hence shows the relative accuracy that may be reasonably expected
in lines of different length. The chief point of interest in the law
lies in the fact that the error in a base line is not likely to increase
any faster than the square root of its length, so that the probable
error where a line is made four times as long should not be more
than doubled, and so on.
Example. A base line measured imder certain conditions has the value
7716.982 ± 0.028 ft. What is the theoretical probable error of a base line
15693.284 ft. long, measured under the same conditions?
0.028 J
16693.284 ^^„_„3gg_
7716.982
Theoretical probable error of new Une = ± 0.0399 ft.
203. The Law of Relative Weight. In accordance with
the law of the previous article, we may write for the probable
error of a base line of any length
rL = mVL, (90)
in which m is a coefficient depending on the conditions of measure
ment. Also in accordance with the law of Art. 172, we may write
1
rz, = s — ^,
Vp
in which p is the weight assigned to the line and s is a coefficient
depending on the unit of weight and the conditions of measure
338 GEODETIC SURVEYING
ment. Since the xinit of weight is entirely arbitrary we may assign
that value to p which will make s equal m, and write
rL==m^ (91)
Combining Eqs. (90) and (91), we have
m\/L = m—i=L;
Vp
from which
p = ^; (92)
whence we have the
General Law: Under the same conditions of measurement the
weight of a base line varies inversely as its length.
From the manner in which this law has been derived it is
evident that it is theoretically true whether the length assigned
to a base line is the result of a single measurement, or the average
of a nvmiber of measurements, provided the lines compared have
all been measured in the same way.
If two or more base lines are measured under different con
ditions, they may be first weighted so as to offset this circum
stance, and then weighted inversely as their lengths. The
relative weight of each line will then be the product of the weights
applied to it.
204. Probable Error of a Line of Unit Length. The probable
error of an angular measurement conveys an absolute idea of its
precision without regard to the size of the angle. The probable
error of a base line, however, conveys no idea of the precision
of the work imless accompanied by the length of the line. It is
therefore convenient to reduce the probable error of a base line
to its corresponding value for a similar line of unit length. A
unit of comparison is thus established for different grades or
pieces of work which is independent of the length of the bases.
Such a unit has no actual existence, but is purely a mathematical
basis of comparison.
From Eq. (89) we have
rL = ^VL.
Vt
APPLICATION TO BASELINE WOEK 339
Hence, when L equals 1, we have for Tq, the probable error of a
unit length of line,
_ ''«
whence in general
tl = roVZ, (93)
in which all the values refer to single measurements. From this
equation we see that the probable error of any base line is equal
to the square root of its length multiplied by the probable error
of a imit length of such a line. If r^ is well determined for
given instruments, conditions, and methods, Eq. (93) informs us
in advance what is a suitable probable error for a single measure
ment, and hence (Art. 198) for the average of any number of
measurements of a line of the given length L. The baseline
party therefore knows whether its work is up to standard, or
whether additional measurements are required.
205. Determination of the Numerical Value of the Probable
Error of a Line of Unit Length, From Eq. (93) we have,
rjr = TqZL;
whence
vi <"'"
So that in any case where the length of a line and the correspond
ing probable error are known, the formula determines a value
for ro. In order for the value of ro to be reliable it must be based
on many such determinations, but the expense prohibits many
measurements of a long base line. As the law is known, however,
which connects the values of the probable error for all lengths
of line, it is just as satisfactory to determine ro from much shorter
lines, which may be quickly and cheaply measured many times.
The usual plan is to measure a series of duplicate lines, so that the
probable error for a single measurement is known in each case
from the discrepancy in each pair of lines. Since all results are
reduced to the same unit length it is immaterial whether the
different duplicate lines are, of equal length or not.
340 GEODETIC SUEVEYING
In accordance with Eq. (84) we have, for any single measure
ment of the duplicate line I,
n = 0.4769\/d2;
whence, in accordance with Eq. (94),
but, in accordance with Eq. (92), we have for any length of line I
1 ■
whence
Vq = QA769Vpd^, (95)
when determined from a single duplicate line. If a number of
duplicate hues are measured we will have a corresponding number
of values {ro)i, (''0)2, etc., based on the discrepancies di, d2, etc.,
of the several duplicate lines. It might as first be supposed that
the average value of these determinations of ro would best repre
sent the result of all the measurements. What is really wanted,
however, is that value of rg which gives equal recognition to the
conditions which caused its different values. A just recognition
of each value of ro, therefore, will require us to consider equal
sections of any line as having been measured respectively under
those conditions that produced the several values of Tq. The
probable error for the whole line is then found from the probable
errors of the different sections, and this result reduced to the
probable error of a unit length.
Let n = the number of values (ro):, (ro)2, etc.;
L = the length of any given line;
whence the required equal sections will be
a = (^) = etc. ^^,
\n/i \nj 2 n
and, in accordance with Eq. (93),
r ^^^ = (ro) lyg, r ^^ = {r,) 2 ^^, etc. ;
APPLICATION TO BASELINE WORK
whence, in accordance with Eq. (88),
and, in accordance with Eq. (94),
""o V~rr'
but, in accordance with Eq. (95),
(ro)i = 0.4769V pd?, (ro)2 = 0.4769 V^"?, etc.;
Sro2 = (0.4769)2Spd2;
341
(96)
so that
whence
rQ = 0.4769
(97)
when determined from a number of duplicate lines. In using
formulas (95) and (97) it is to be remembered that d is the dis
crepancy in any duplicate line, p is the weight (reciprocal of the
length) of that line, n is the number of duplicate lines, and ro
is the probable error of a single measurement of a line of unit
length.
Example. Determination and application of the probable error of a base
line of unit length :
Duplicate Lines
512.017 ft.
512.011 "
d
0.006
d2
0.000036
p
0.0000000703
619.184 ft.
619.176 "
0.008
0.000064
eio
0.0000001034
750.962 ft.
750.971 "
0.009
0.000081
Th
0.0000001079
619.180 ft.
619.184 "
0.004
0.000016
619
0.0000000258
750.960 ft.
750.972 "
0.012
0.000144
1
761
0.0000001917
from which we have
S
whence
0.4
= 0.0000004991 <
md n
= ± O.OC
= 5;
To =
^cn../ 0.0000004991
769'V
)0151 ft.,
342 GEODETIC SURVEYING
which is therefore the probable error for a single measurement of one foot
made under the given conditions. For a single measurement of a base
line of any length L, therefore, made under these same conditions, the
probable error would be, in accordance with Eq. (93),
Tl = 70 Vl= ± 0.000151 Vl ft.
Thus if L is 10,000 feet, we would have
ri. = ± 0.000151 X VlOOOO = ± 0.0151 ft.
And if such a line were measured four times we should have, theoretically, for
the probable error of the average length,
ra= ± 0.0151 r vT = ± 0.0076 ft.
It thus becomes known in advance what probable error is to be expected
under the given conditions.
206. The Uncertainty of ^ Base Line. By the uncertainty
of a base line is meant the value obtained by dividing its probable
error by its length. In accordance with Art. 202, the probable
error of a base line varies as the square root of its length, so that
the probable error increases much more slowly than the length
of the line. On account of the greater opportunity for the
compensation of errors, therefore, long lines are relatively more
accurate than short lines. While the unit probable error r^
very satisfactorily indicates the grade of accuracy, whether a
line be long or short, it does not furnish any idea of the degree of
accuracy with which the length of a given line is known. The
uncertainty of a base line, however, shows at once the precision
attained in its measurement. If ri be the probable error of a
single measurement of a base line whose length is I, then for the
uncertainty C/j of a single measurement, we have
and for the uncertainty U^ of the arithmetic mean of n measure
ments,
f/„ = r? = _!i_.
But, in accordance with Eq. (93),
n = roVT;
APPLICATION TO BASELINE WORK 343
whence
and
so that we may write,
and
_ro\/r_ rp '
' I VJ'
IV n Vrii
^^=l = Vf' (^^)
C/„ = ^=^ (99)
' Vnl
Example 1. Three measurements of a base line under the same con
ditions give z = 6716.626 ± 0.0088 ft. and n = ± 0.0152 ft. What is the
uncertainty of a single measurement and also of the arithmetic mean?
rr Ti 0.0152 1
Ua
I 6717.626 441949'
ra ^ 0.0088 ^ 1
I 6717.626 763366'
Example 2. A base line of 10,000 ft. length is to be measured four times
under conditions which make the probable error of a unit length of line
equal ± 0.000316 ft. What should be the uncertainty of each measurement
and of the average of the four measurements?
„ To 0.000316 1
fi =
Ua =
VT VlOOOO 316456'
To 0.000316 1
vW V 40000 632912
CHAPTER XVI
APPLICATION TO LEVEL WORK
207. Unweighted Meastirements. If the difference of ele
vation of two stations is msiasured a number of times in the same
manner, over the same length of line, and under such conditions
that the different determinations may be regarded as of equal
weight, then (Art. 155) the arithmetic mean of the several results
is the most probable value of this difference of elevation. The
probable error of a single measurement (Art. 173) is given by the
formula
ri = 0.6745 J;^73^, (100)
and the probable error of the arithmetic mean (Art. 173) of n
measurements by the formula
■a = ~ = 0.674:5 J .^"^ ,, . . . . (101)
i(n  1)' ■ ■
Example. Difference of elevation by direct observations of equal weight
Observed Values
V
12
11.501ft.
+ 0.009
0.000081
11.509 ft.
+ 0.017
0.000289
11.480 ft.
 0.012
0.000144
11.478 ft.
0.014
0.000196
4)45.968 ft.
Sd^ = 0.000710
= 11.492 ft.
n = 4
n = 0.6745 J'
'00007J0^^„^„^^^^^_
., = 0^4 ^±0.0052 ft.
\/4
Most probable value = 11.492 ± 0.0052 ft.
344
APPLICATION TO LEVEL WOEK 345
208. Weighted Measurements. If the difference of eleva
tion of two stations is measured a number of times in the same
manner, and over the same length of line, but under such condi
tions that the different determinations must be regarded as of
unequal weight, then (Art. 157) the weighted arithmetic mean of
the several results is the most probable value of this difference of
elevation. The probable error of a single measurement of unit
weight (Art. 174) is given by the formula
n = 0.6745^^^, (102)
the probable error of any measurement of the weight p (Art. 174)
by the formula
r, = ^ = 0.6745 /_^F!!^, .... ao3)
vp \ pin — 1)
and the probable error of the weighted arithmetic mean (Art. 174)
by the formula
;^ = 0.6745 J^.^^. . . . (104)
VSp ■ \ ^Pin  1)
Example. Difference of elevation by direct observations of unequal
weight:
Observed Values p pM v v^ pv^
17.643 ft. 1 17.643 0.028 0.000784 0.000784
17.647 ft. 1 17.647 0.024 0.000576 0.000576
17 679 ft. 2 35.358 +0.008 0.000064 0.000128
17.683 ft. 3 53.049 +0.012 0.00 0144 0.000432
Sp = 7 ) 123.697. tpv^ = 0.001920
z = 17.671 n = 4
^4
n = 0.6745. /^^^5^5?2. = ± 0.0171 ft.
r, = '':^=±Qm2Ut.
n =
^•na
V2
0.0171
0.0171
= ± 0.0099 ft.
= ± 0.0064 ft.
Most probable value = 17.671 ± 0.0064 ft.
346 GEODETIC SUEVEYING
209. Duplicate Lines. In precise level work a duplicate line
of levels is understood to mean a line which is run twice over the
same route with equal care, but in opposite directions. The
object of running in opposite directions is to eliminate from the
mean result those systematic errors which are liable to occur in
leveling, due to a risiag or settling of the instrument or tiuning
points during the progress of the work. As explained in Art. 88
the details of the work are so arranged that these errors tend to
neutralize each other to a large extent as the work progresses, so
that no material error is committed by assuming that the results
obtained are affected only by accidental errors. The most prob
able value for the difference of elevation of any two stations,
based on a duplicate line, is equal to the average of the two results
furnished by such a line. Letting d represent the discrepancy
between the result obtained from the forward line and that
obtained from the reverse line, we thus have
d , d
vi = +2 and V2 =  ^.
Substituting these values in Eq. (100) and replacing n with r,
for the case of duplicate lines, we have for the probable error
of a single determination (forward or reverse) by a line of the
length I,
ri = 0.4769\/d2 = 0A769d (105)
Substituting the same values in Eq. (101), we have for the
probable error of the arithmetic mean of the results obtained by
the forward and reverse lines,
whence
r„ = 0.3348d; (106)
Ta (approximately) = id (107)
Example. Duplicate liiie of levels:
Observed Values
29.648 ft. 0.4769 X 0.028 = 0.0134.
29.676 ft. 0.3348 X 0.028 = 0.0094.
d = 0.028 ft.
r; = ± 0.0134 ft. Ta = ± 0.0094 ft.
Most probable value = 29.662 ± 0.0094 ft.
APPLICATION TO LEVEL WORK 347
210. Sectional Lines. Every line of levels which includes
one or more intermediate bench marks may be regarded as made
up of a series of sections connecting these bench marks. In
general the work will be done by the method of duplicate leveling
(Art. 209), so that a value for the difference of elevation of any two
successive bench marks (limiting a section) will be obtained from
the forward line, and another value from the reverse line. From
these two values (Art. 209) we will have a most probable value
and a probable error for any given section, which will be independ
ent of all other sections. In whatever manner the leveling may
be done, however, the subsequent treatment of the results will be
the same, provided the determinations for each section are kept
independent. If ei, 62, . . . en, be the most probable values for
the difference of elevation between the successive bench marks,
then (Art. 168) the most probable difference of elevation E
between the terminal bench marks, is
^ = ei + 62 . . . +en = 2e. . . . (108)
And if ri, r2, . . . Vn, be the probable errors of the several values
ei, 62, etc., then (Art. 182) the probable error r^j for the total dif
ference of elevation E, is
rE
Vri2 + ra^ . . . + r„2 = vTr^. . . . (109)
Example. Level work on sectional lines. Given
ei = 9.116 ± 0.008 ft.
62 = 31.659 ± 0.031 ft.
63 = 22.427 ± 0.018 ft.
E = 9.116 + 31.659 + 22.427 = 63.202 ft.
r^ = V(0.008)2 + (0.031)'! + (o.018y = ± 0.037 ft.
Most probable value E = 63.202 ± 0.037 ft.
211. General Law of the Probable Errors. In measuring
the difference of elevation between any two bench marks by pass
ing (in the usual way) through a series of turning points, the case
is essentially one of sectional measurement (Art. 210), in which the
difference of elevation for each section is measured a single time,
and in which under similar conditions the average distance
between turning points may be assumed to be the same for any
length of line. Running a line of levels is thus entirely analogous
348 GEODETIC SURVEYING
to measuring a base line, and hence the same laws must hold good.
In accordance with Art. 202, and without further demonstration,
we may therefore write as a
General Law: Under the same conditions of measurement
the probable error of a line of levels varies as the square root of its
length.
From the considerations on which this law is based it is evident
that it is theoretically true whether the difference of elevation
assigned to the terminals of a line is the result of a single measure
ment, a number of measurements, or a duplicate measurement, so
long as the lines being compared are all identical in these details.
Example. A line of levels 10 miles long has a probable error of ± 0.156 ft.
What is the theoretical value of the probable error for a Mne 60 miles long,
run under the same conditions?
0.156 V5 = 0.156 V6"= ± 0.382 ft.
Theoretical probable error of new line = ± 0.382 ft.
212. The Law of Relative Weight. As explained in the
previous article, the laws derived for baseline work are equally
applicable to level work. In accordance with Art. 203, and with
out further demonstration, we may therefore write as a
General Law : Under the same conditions of measurement the
weight of the result due to any line of levels varies inversely as the
length of the line.
From the considerations on which this law is based it is evident
that it is theoretically true whether the difference of elevation
assigned to the terminals of the line is the result of a single meas
urement, a number of measurements, or a duplicate measurement,
so long as the lines being compared are all identical in these
details.
If two or more level lines are run under different conditions,
they may be first weighted so as to offset this circumstance, and
then weighted inversely as their lengths. The relative weight of
each line will then be the product of the weights applied to it.
213. Probable Error of a Line of Unit Length. The probable
error corresponding to a given line of levels conveys no idea of the
precision of the work unless accompanied by the length of the line.
It is therefore convenient to reduce the probable error of a line of
levels to its corresponding value for a similar line of unit length.
APPLICATION TO LEVEL WORK 349
A unit of comparison is thus established for different grades or
pieces of work which is independent of the length of the lines.
Such a unit has no actual existence, but is purely a mathematical
basis of comparison.
As explained in Art. 211, the laws derived for baseline work
are equally applicable to level work. In accordance with Art. 204,
and without further demonstration, we may therefore write
r^ = ?oVL, (110)
in which tl is the probable error for a given line of levels of the
length L, Vq is the probable error for a unit length of such a line,
and in which all the values refer to single measurements. This
equation indicates that the probable error of any given line of
levels is equal to the square root of its length multiplied by the
probable error for a unit length of such a line. If Tq is well deter
mined for given instruments, conditions, and methods, Eq. (110)
informs us in advance what is a suitable probable error for a
single line of levels, and hence (Art. 207) for the average result
obtained by rerunning such a line any number of times. In
accordance with this article the probable error in the mean result
of a duplicate line is equal to the second member of Eq. (110)
divided by V2. In any case, therefore, the level party knows
whether its work is up to standard, or whether additional measure
ments are required.
214. Determination of the Numerical Value of the Probable
Error of a Line of Unit Length. As explained in Art. 211, the
laws and rules for baseline work are equally applicable to level
work. The method of Art. 205 is consequently adapted to the
present case by running ofte or more duplicate level lines of
moderate length, and noting the length of line (one way) and the
discrepancy for each duplicate line. In accordance with Eq.(97),
and without further demonstration, we may therefore write
0.4769
\IS., (Ill)
\ n
in which Tq is the probable error in running a single line of levels
of unit length, d is the discrepancy in any duplicate line, p is
the weight (reciprocal of the one way length) of that line, and n
is the number of duplicate lines.
350
GEODETIC SURVEYING
Example. Determination and application of the probable error of a
level line of unit length:
Difference of Elevation d
d'
I
p
pd2
16.298 ft.
16.314"
0.016
0.000256
810
^
0.0000003160
16.308 ft.
16.296"
0.012
0.000144
810
8^0
0.0000001778
18.540 ft.
18.549"
0.009
0.000081
560
1
560
0.0000001446
18.552 ft
18.542"
0.010
0.000100
560
beo
0.0000001786
21.663 ft.
. 21.648"
0.015
0.000225
782
ih
0.0000003085
21.661ft.
21 649 "
0.012
0.000144
782
^
0.0000001841
21.664 ft.
21.650"
0.014
0.000196
782
Th
0.0000002506
from which we have
whence
Xpd" = 0.0000015602 and n = 7;
r„ = 0.4769
V°=
1.0000015602
= ± 0.000225 ft.,
which is therefore the probable error in running a single line of levels for
a distance of one foot under the given conditions. For a single line of levels
of any length L, run under the same conditions, the probable error would
be, in accordance with Eq. (110),
TL =roVL = ± 0.000225^1" ft.
Thus if L is 10,000 feet, we would have
Tl = ± 0.000225^/10000 = ± 0.0225 ft.
And if such a line of levels were run four successive times we should have,
theoretically, for the probable error of the average difference of elevation,
ra = ± 0.0225 ^ Vi" = ± 0.0113 ft.
It thus becomes known in advance what probable error is to be expected
under the given conditions.
215. Mtiltiple Lines. By a multiple line of levels is meant a
set of two or more lines connecting the same two bench marks
by routes of different length. In order to find the most probable
value for the difference of elevation between the terminals of a
multiple line, it is neces.sary (Art. 212) to weight each constituent
line inversely at its length. If the character of the work requires
any of the lines to be also weighted for other causes, then the
APPLICATION TO LEVEL WOEK 351
final weight of such hne must be taken as the product of its indi
vidual weights. Having weighted the several lines as thus explained
the case becomes identical with any case of weighted measure
ments (Art. 208), and hence the probable error of a single measure
ment of unit weight is given by the formula
'•i
= '''^'yl^'' ^112)
the probable error of any of the lines of the weight p by the
formula
and the probable error of the weighted arithmetic mean by the
formula
^^ = 0.6745 Jy^^y .
V:^p MZpin  1)
V = ^1= = 0.6745 V ^^,: _,, . . . (114)
._ 5 Miles
.2i^ Miles.
~ 3% MUes 
Fig. 86.
Example. Three lines of levels, as shown in Fig. 86, give the following
results :
A to B, 5 mile line, + 95.659 ft.
A to B, 2i mile Hne, + 95.814 ft.
A to B, 3i mile Hne, + 95.867 ft.
The elevation of A is 416.723 feet. What is the most probable value for
the elevation of B, and the probable error of this result?
M
95.659
95.814
95.867
p pM
0.2 19.1318
0.4 38.3256
0.3 28.7601
V
 0.138
+ 0.017
+ 0.070
»2
0.019044
0.000289
0.004900
pu2
0.0038088
0.0001156
0.0014700
Sp
= 0.9J86.2175
95.797
rpa = 0.67451
n =
± 0.0369 ft.
: 0.0053944
3
/0.OO53944 _
416.723 + 95.797 = 512.520 ft.
Most probable value for elevation of B = 512.520 ± 0.0369 ft.
352
GEODETIC SURVEYING
216. Level Nets. When three or more bench marks are
interconnected by level lines so as to form a combination of
closed rings, the resulting figure is called a level net. Fig. 87
represents such a level net, involving nine bench marks. The
elevation of any bench mark is necessarily independent of any
other bench mark, but the differences between the elevations of
adjacent bench marks are not independent quantities, since in
any closed circuit their algebraic sum must equal zero. In the
given figure there are evidently fifteen observation equations,
namely, the observed difference of elevation between A and B,
B and C, etc. But there are also seven closed rings, ABCD, ADA,
etc., forming seven independent condi
tional equations. Fifteen minus seven
leaves eight, so that (Art. 166) there
can be but eight independent quanti
ties involved in the fifteen observation
equations. The number of indepen
dent quantities must evidently be one
less than the number of bench marks,
since one of these must be assumed as
known or fixed, and nine minus one
gives eight as before. It sometimes
happens that more than one line con
nects the same two points, as between
A and D in the fi ure; but this fact
makes no difference in the method of
computation. Sometimes a point B
occurs on a line without being coimected with any other point.
Such a point has no influence on the adjustments of any other
point, and may be included or omitted, as preferred, in making
such other adjustments. If omitted ^in adjusting the other
points its own most probable value can be found afterwards
by Art. 217.
There are two general methods of making the computations
for the adjustments of a level net, each of which may be modified
in a number of ways. In the first method the most probable
values are found for the several differences of elevation between
the bench marks, the most probable values for the elevations of
the different bench marks being then found hy combining these
differences. In the second method the computations are arranged so
APPLICATION TO LEVEL WOEK
353
as to lead directly to the most probable values for the elevations
of the bench marks. In any case each of the connecting lines
must be properly weighted. If the lines are all run singly they
are weighted inversely as their lengths unless some special con
dition requires some of these weights to be modified. If all the
lines are duplicate lines, the average difference of elevation in
each case may be treated as if due to a single line, and weighted
inversely as its length. If special conditions exist the weights
must be made to correspond. The manner
in which each method is worked out is
illustrated by the following example.
Example. Referring to the level net indicated
in Fig, 88, the field notes show the following
results:
AtoB = + 11.841 ft.
= +
Bto C
C toD
Dto E = 
EtoA
B to E = 
C toE = +
5.496 ft.
8.207 ft.
6.720 ft.
8.515 ft.
3.218 ft.
2.619 ft.
The figures on the diagram are the lengths in miles
of the various lines. The arrowheads show the
direction in which each hne was run. The eleva
tion of the point A is 610.693 ft. What are the
most probable values for the elevations of the re
maining stations?
First method. As there are but four unknown
bench marks (5, C, D, E), there can be but four in
dependent unknowns in the observation equations.
As the lines AB, BC, CD, DE, may evidently be selected as the independent
unknowns, we may write for the most probable values of the corresponding
differences of elevation
AtoB=+ 11.841 + vi;
B toC =  5.496 4 W,
C toD= + 8.207 + V3;
DtoE =  5.720 + U4.
The conditional equations involved in the several closed circuits may then
be avoided (Art. 165) by writing all the observation equations in terms of
these quantities. Writing the reduced observation equations (Art. 163)
directly from the figure, we have, by comparison with the observed values,
(A to B) vi = 0.000 (weight 0.4)
{B to C) Vi = 0.000 (weight 0.3)
(C to D) Vi = 0.000 (weight 0.4)
(D to E) Vi = 0.000 (weight 0.3)
\e to A) V  vi  V3  Vi = + 0.317 (weight 0.2)
(B to E) V2 + V3 + Vi=  0.209 (weight 0.5)
(C to E) V3+Vi = + 0.132 (weight 0.5)
354 GEODETIC SURVEYING
As an illustration of how these equations are formed let us consider the
observed line CE.
Most probable value, C to D = + 8.207 + Vt.
Most probable value, Dto E =  5.720 + Vt.
Hence, by addition,
Most probable value, C to ^ = + 2.487 + % + Vi.
Observed value, C to E = + 2.619.
Hence this observation equation requires
Vi+Vi = + 0.132.
No values of Vi, %, Vs, vt, can meet the requirements of all the observation
equations, and hence to find the most probable values of Vi, v^, Va, vt, we
form the normal equations in the usual way, giving,
0.6wi + 0.2!;2 + 0.2W3 + 0.2^4 =  0.0634
0.2iii + l.Owj + 0.7w3 + 0.7u4 =  0.1679
0.2i;i + 0.7v2 + 1.6ws + 1.2w4 =  0.1019
0.2di + 0.7w2 + 1.2 vs+ 1.5K4 =  0.1019
whose solution gives
vi =  0.0556 ft.; «3 = + 0.0092 ft.;
Vi=  0.1718 ft.; vi = + 0.0123 ft.;
whence, for the most probable values, we have
AtoB = + 11.7854 ft.
B to C =  5.6678 " A = 610.693 ft.
C to D = + 8.2162 " B = 622.478 "
DtoE= 5.7077" 0=616.811"
EtoA =  8.6261 " D = 625.027 "
BtoE= 3.1593" .E = 619.319"
CtoE = + 2.5085"
Second method. In this method we first find approximate values for the
unknown elevations by combining the observed values in any convenient
way, thus:
A = 610.693 C = 617.038 (approx.)
+ 11.841 + 8.207
B = 622.534 (approx.) D = 625.245 (approx.)
 5.496  5.720
C = 617.038 (approx.) E = 619.525 (approx;)
and then write, for the most probable values,
A = 610.693;
B = 622.534 + Vi;
C = 617.038 + t)2;
D = 625.245 + us;
E = 619.525 + Vi.
APPLICATION TO LEVEL WOEK
355
Substituting these values in the observation equations, we have
Ato B = + 11.841 +vi = + 11.841:
BtoC = 
5.496
«! + %= 
5.496
C to D = +
8.207
 f 2 + Va = +
8.207
DtoE = 
6.720
— Va + Vi = —
5.720
E toA = 
8.832
Vi = 
8.515
BtoE = 
3.009
Vi+Vi= 
3.218
CtoE = +
2.487
 % + Wd = +
2.619.
Eeducing and weighting inversely as the distances, we have
vi = 0.000 (weight 0.4)
vi + Vi = 0.000 (weight 0.3)
vi + vi = 0.000 (weight 0.4)
V3 + Vi= 0.000 (weight 0.3)
 v,= + 0.317 (weight 0.2)
Vi +Vi =  0.209 (weight 0.5)
V2 +Vi = + 0.312 (weight 0.5)
Forming the normal equations, we have
1.2j)i  O.Sfz  0.5w4 = + 0.1045
 0.3i;i + 1.2i;2  OAvz  0.5t)4 =  0.0660
 0Av2 + 0.7v3  0.3w4 = 0.0000
 0.5vi  0.5z)2  0.3% + 1.5w4 =  0.1019
whose solution gives
11 =  0.0566 ft.; V3 =  0.2182 ft.;
V2=  0.2274 " Vi=  0.2069 "
whence, for the most probable values, we have (as before)
A = 610.693 ft.
B = 622.478"
C = 616.811"
D = 625.027"
E = 619.319"
217. Intermediate Points. By an inter
mediate point is meant one lying only on
a single line of levels, and hence having
n9 influence on the general adjustment.
Thus in Fig. 89 the bench marks A and B
are adjusted as a part of the complete level
net ABCDEFG. The point I is an inter
mediate point, having no influence on the
general adjustment, but simply lying be
tween the djusted bench marks A and B.
In adjusting level net it s not necessary
to separate the intermediate points from the others, as the
results will come out the same whether any or all of the inter
mediate points are omitted or included. The work of compu
FiG. 89.
356 GEODETIC SURVEYING
tation may be reduced, however, where there are many inter
mediate points, by adjusting the main system first and the inter
mediate points afterwards. Referring to Fig. 89, page 355,
Let I be an intermediate point lying between the adjusted
bench marks A and B;
a = the distance A to 7;
b = the distance I to B;
d = the discrepancy between the line AB as run and the
difference between the adjusted values of A and B
(+ if the line as run makes B too high) ;
e = observed change in elevation from A to I;
e' = observed change in elevation from I to B;
then
A+e + e' = B + d,
or
and
e' = B  A  e + d;
I (observed) = A + e (weight b) ;
I (observed) =B — e' = A+e — d (weight a);
or, taking the weighted arithmetic mean,
bA + be + aA + ae — ad
b + a
I (most probable)
As / represents any intermediate point, and a the corresponding
distance from the commencement A of the given
line, it follows from this equation that the most
probable values for any intermediate points are
U Miles, arrived at by adjusting for the discrepancy d in
direct proportion to the distances from the initial
point A. This law may be otherwise expressed
by saying that the discrepancy is to be distributed
uniformly along the line on the basis of dis
^^™«^:, tance.
Example. In the line of levels indicated in Fig. 90 the
field notes show the following changes in elevation:
V 2 Miles.
AtoB = + 2.626 ft.
^A BtoC =  3.483"
Fig. 90. C to D = +6.915"
APPLICATION TO LEVEL WOEK
357
The adjusted elevations at A and D are
A = 28.655 ft.
D = 34.317"
What are the most probable elevations of the intermediate points B and C?
28.655
+ 2.626
Discrepancy = + 0.396 ft. Total distance = 9 miles.
31.281 0.396 X f = 0.088 ft. 0.396 X  = 0.220 ft.
 3.483
27.798
+ 6.915
34.713
34.317
ition
Apparent Elevation
Correction
Adjusted Elevation
A
28.655
0.000
28.655 ft.
B
31.281
 0.088
31.193"
C
27.798
 0.220
27.578"
D
34.713
 0.396
34.317"
+ 0.396
218. Closed Circuits. By a closed circuit in level work is
meant a line of levels which returns to the initial point, or, in
other words, forms a single closed ring. The shape of such a circuit
is entirely immaterial, whether approxi
mately circular, narrow and elongated,
or irregular in any degree. A level net
is in general a combination of closed
circuits, but these circuits can not be
adjusted separately, as they are not
independent. So also if any part of
the ring is leveled over more than once
it becomes essentially a level net, and
must be adjusted accordingly. If, how
ever, the circuit is independent of all
other work, and has been run around but once under uniform
conditions, it may be adjusted by a simpler process. Referring
to Fig. 91,
Let A, B, C, D, E be the bench marks on an independent
closed circuit;
A = the initial bench mark;
a = distance ABC to any point C;
b = distance CDEA back to A ;
d =1= discrepancy on arriving at A ( + if too high) ;
Fig. 91.
then
e = observed change in elevation from A to C;
e' = observed change in elevation from C to A;
A + e + e' = A + d,
358
GEODETIC SURVEYING
or
and
e'= — e + d;
C (observed) = A + e (weight b) ;
C (observed) = A —e' = A + e — d (weight a);
or, taking the weighted arithmetic mean,
bA + be + aA + ae — ad
C (most probable)
= {A + e)
b + a
a
a + b
(116)
As C represents any point in the circuit, and a the corresponding
distance from the initial point A, it follows from this equation
that the most probable values for the elevations of any points
B, C, D, E, etc., are arrived at by adjusting the observed eleva
tions for the discrepancy d directly as the respective distances
from the initial point. This law may be otherwise expressed by
saying that the discrepancy is to be distributed uniformly around
the circuit on the basis of distance.
Example. In the closed line of levels indicated in Fig. 91, page 357, the
field notes show the following changes in elevation:
AtoB =  2.176 ft.,
BtoC=+ 6.481 ft.,
C to D ^ 1.712 ft.,
DtoE =  4.820 ft.,
EtoA = + 2.017 ft..
Given the elevation of A as 47.913 feet, what are the adjusted elevations
around the line?
47.913
 2.176
distance = 3 miles,
distance = 1 mile,
distance = 2 miles,
distance = 2 miles,
distance = 3 miles.
45.737
+ 6.481
52.218
 1.712
50.506
 4.820
45.686
+ 2.017
47.703
47.913
Discrepancy = — 0.210 ft. Total distance = 11 miles.
0.210 X A = 0.057 ft 0.210 X t\= 0.105 ft.
0.210 X A = 0.076 ft. 0.210X t\= 0.153 ft.
Station Apparent Elevation Correction Adjusted Elevation
A
47.913
0.000
47.913 ft.
B
45.737
+ 0.057
45.794"
C
52.218
+ 0.076
52.294 "
D
60.506
+ 0.105
50.611 "
E
45.686
+ 0.153
45.839"
0.210
APPLICATION TO LEVEL WOEK
359
219. Branch Lines, Circuits, and Nets. Any level line, circuit,
or net that is independent of another
system except for one common point,
is called a branch system. Thus in
Fig. 92 the dotted lines represent the
original system, ABCD a branch line,
HKLMN a branch circuit, and PRSTV
a branch net. In adjusting the main
system the results will be the same
whether any or all of the branch sys
tems are included or omitted. If
there is much branch work, however,
the labor of computation may be re
duced by adjusting the main system
first and the branch systems after
wards. When the main system is
adjusted the elevations ofA,H, P, etc.,
become fixed quantities which must not be disturbed in adjusting
the branch systems.
FiQ. 92.
TABLES
TABLES
TABLE I.— CURVATURE AND REFRACTION (IN ELEVATION)*
Difference in Feet for
Difference in Feet for
Dis
tance,
Dis
tance.
Milea.
Curvature.
Refraction.
Curvature
and
Refraction.
Miles.
Curvature.
RSraction.
Curvature
and
Refraction.
1
0.7
0.1
0.6
34
771.3
108.0
663.3
2
2.7
0.4
2.3
35
817.4
114.4
703.0
3
6.0
0.8
5.2
36
864.8
121.1
743.7
4
10.7
1.5
9.2
37
913.5
127.9
785.6
5
16.7
2.3
14.4
38
963.5
134.9
828.6
6
24.0
3.4
20.6
39
1014.9
142.1
872.8
7
32.7
4.6
28.1
40
1067.6
149.5
918.1
8
42.7
6.0
36.7
41
1121.7
157.0
964.7
9
54.0
7.6
46.4
42
1177.0
164.8
1012.2
10
66.7
9.3
57.4
43
1233.7
172.7
1061.0
11
80.7
11.3
69.4
44
1291.8
180.8
1111.0
12
96.1
13.4
82.7
45
1351.2
189.2
1162.0
13
112.8
15.8
97.0
46
1411.9
197.7
1214.2
14
130.8
18.3
112.5
47
1474.0
206.3
1267.7
15
150.1
21.0
129.1
48
1537.3
215.2
1322.1
16
170.8
23.9
146.9
49
1602.0
224.3
1377.7
17
192.8
27.0
165.8
50
1668.1
233.6
1434.6
18
216.2
30.3
185.9
51
1735.5
243.0
1492.5
19
240.9
33.7
207,2
52
1804.2
252.6
1551.6
20
266.9
37.4
229.5
53
1874.3
262.4
1611.9
21
294.3
41.2
253.1
54
1945.7
272.4
1673.3
22
322.9
45.2
277.7
55
2018.4
282.6
1735.8
23
353.0
49.4
303.6
56
2092.5
292.9
1799.6
24
384.3
53.8
330.5
67
2167.9
303.6
1864.4
25
417.0
58.4
358.6
58
2244.6
314.2
1930.4
26
451.1
63.1
388.0
59
2322.7
325.2
1997.6
27
486.4
68.1
418.3
60
2402.1
336.3
2065.8
28
523.1
73.2
449.9
61
2482.8
347.6
2135.2
29
561.2
78.6
482.6
62
2564.9
359.1
2205.8
30
600.5
84.1
516.4
63
2648.3
370.8
2277.5
31
641.2
89.8
551.4
64
2733.0
382.6
2350.4
32
683.3
95.7
587.6
65
2819.1
394.7
2424.4
33
726.6
101.7
624.9
66
2906.5
406.9
2499.6
* From Appendix No. 9, Report for 1882, United States Coast and Geodetic Survey.
363
364
GEODETIC SURVEYING
TABLE II.— LOGARITHMS OF THE PUISSANT FACTORS*
(In U. S. Legal Meters)
Lat.
A
B
C
D
E
F
o
10
 10
 10
 10
— 20
— 20
20
8.5095499
8.512155s
0.96732
2 . 1996
5.7574
7.772
21
8 • 5095330
8.5121049
0.99036
2.2170
5. 77"
7.787
22
8.5095155
8.5120524
I. 01252
2 . 2333
57851
7.800
23
8 . 5094973
8.5119979
1.03389
2.2485
5.7997
7.812
24
8.5094786
8.5119416
1.05455
2.2627
5. 8146
7.823
25
8 . 5094592
8. 51 18834
1.07456
2.2759
5.8300
7832
26
8.5094392
8,5118236
1.09399
2.2882
5 . 8458
7.841.
27
8 . 5094187
8.5117620
1.11289
2.2997
5 . 8620
7.849)
28
8.5093977
8.5116989
1.13131
2.3104
5.8785
7.855
29
8.5093761
8.5116342
114931
2.3203
5.8955
7.861
30
8.5093,541
8,5115682
1.16691
2.3294
5.9127
7.866
31
8.5093316
8,5115007
1.18415
2.3379
5.9304
7.870
32
8 . 5093087
8.5114321
I. 20107
2.3456
5 9484
7.873
33
8 . 5092854
8.5113622
1.21771
2.3527
5.9667
7.87s
34
8.5092618
8.5112912
I . 23408
2.3592
5 9853
7.877
35
8.5092378
8.5112192
1.25023
2.3651
6.0043
7.877
36
8.5092135
8.5111463
I. 26616
2.3704
6.0237
7.877
37
8.5091889
8. 51 10725
I .28192
2,3750
6.0433
7.876
38
8.5091640
8.5109980
1,29752
2.3792
6.0633
7.874
39
8.5091390
8.5109228
I. 31298
2.3827
6.0836
7.872
40
8. 5091 137
8.5108470
1.32832
2,3857
6.1043
7.869
41
8 . 5090883
8.5107708
I. 34357
2.3882
6.1253
7.864
42
8.5090628
8.5106942
1.35874
2.3901
6.1467
7.860
43
8.5090372
8.5106173
1,37385
2,3914
6.1684
7.854
44
8. 50901 I 5
8 . 5105402
1.38893
2.3923
6.1905
7.848
4S
8.5089857
8.5104630
1,40399
2.3926
6.2130
7.840
46
8.5089600
8.5103858
I. 41905
2,3924
6.2359
7.832
47
8 . SO89343
8.5103087
I. 43413
2.3917
6.2592
7.824
48
8 . 5089086
8.5102317
1.44925
2.3904
6 . 2830
7.814
49
8.5088831
8.5101551
1.46442
2.3886
6.3071
7.804
SO
8.5088576
8.5100788
1.47967
2.3862
6.3318
7.792
51
8.5088324
8.5100029
I. 49501
2.3833
6.3569
7.780
52
8.5088073
8 . 5099276
I. 51047
2.3799
6.3826
7.767
53
8.5087824
8.5098530
1.52607
2.3759
6 . 4088
7. 753
54
8.5087577
8.5097791
I. 54182
2.3713
6.4355
7.738
55
8.5087334
8 .5097060
1.55776
2.3661
6 . 4629
7.723
56
8.5087093
8.5096338
1.57390
2 . 3603
6.4909
7.706
57
8.5086856
8 . 5095626
1.59027
2.3539
6.5196
7.688
58
8.5086622
8 . 5094925
I. 6069 I
2.3469
6.5490
7.669
59
8 . S086393
8.5094236
1.62383
2.3392
6.5792
7.649
60
8.5086167
8.5093560
I. 64108
2.3309
6.6102
7.627
61
8 . 5085946
8.5092897
1.65868
2.3218
6.6422
7.605
62
8.5085730
8 . 5092248
1.67667
2.3120
6.6750
7.581
63
8.5085519
8.5091614
1.69509
2.3014
6.7089
7.556
64
8.5085313
8.5090996
I. 71399
2.2901
6.7440
7.529
65
8.5085112
8.5090395
1 ■ 73342
2.2778
6.7802
7.501
66
8.5084917
8. 508981 I
I ■ 75343
2.2647
6.8177
7.471
67
8.5084729
8 . 5089245
1.77409
2.2506
6.8567
7.440
68
8.5084546
8.5088698
I . 79546
2.2354
6.8972
7,406
69
8 . 5084370
8.5088170
I. 81762
2.2192
6,9395
7.371
* Based on tables in App. No. 9, Report for 1894, U. S. Coast and Geodetic Survey.
TABLES
365
TABLE II.LOGARITHMS OF THE PUISSANT FACTORS—
(Continued)
Log G=log diff. for (log 4^)— log diff. for (log s)
log s
log difference.
log JX
logs
log difference.
log JX
3.876
0.0000001
2. 385
4.922
0.0000124
3.431
4.026
002
2.535
4.932
130
3.441
4.114
003
2.623
4.941
136
3.450
4.177
004
2.686
4.950
142
3.459
4.225
005
2.734
4.959
147
3.468
4.265
006
2.774
4.968
153
3.477
4.298
007
2.807
4.976
160
3.485
4.327
008
2.836
4.985
166
3.494
4.353
009
2.862
4.993
172
3.502
4.376
010
2.885
5.002
179
3.511
4.396
Oil
2.905
5.010
186
3.519
4.415
012
2.924
5.017
192
3.526
4.433
013
2.942
5.025
199
3.534
4.449
014
2.958
5.033
206
3.542
4.464
015
2.973
5.040
213
3.549
4.478
016
2.987
5.047
221
3.556
4.491
017
3.000
5.054
228
3.563
4.503
' 018
3.012
5.062
236
3.571
4.526
020
3.035
5.068
243
3.577 '
4.548
023
3.057
5.075
251
3.584
4.570
025
3.079
5.082
259
3.591
4.591
027
3.100
5.088
267
3.597
4.612
030
3.121
5.095
275
3.601
4.631
033
3.140
5.102
284
3.611
4.649
036
3.158
5.108
292
3.617
4.667
039
3.176
5.114
300
3 . 623
4.684
042
3.193
5.120
309
3.629
4.701
045
3.210
5.126
318
3.635
4.716
048
3.225
5.132
327
3.641
4.732
052
3.241
5.138
336
3.647
4.746
056
3.255
5.144
345
3.653
4.761
059
3.270
5.150
354
3.659
4.774
063
3.283
5.156
364
3.665
4.788
067
3.297
5.161
373
3.670
4.801
071
3.310
5.167
383
3.676
4.813
075
3.322
5.172
392
3.681
4.825
080
3.334
5.178
402
3.687
4.834
084
3.343
5.183
412
3.692
4.849
089
3.358
5.188
422
3.697
4.860
094
3.369
5.193
433
3.702
4.871
098
3.380
5.199
443
3.708
4.882
103
3.391
5.204
453
3.713
4.892
108
3.401
5.209
464
3.718
4.903
114
3.412
5.214
474
3.723
4.913
119
3.422
5.219
486
3.728
Note. — The logarithms in the above table require s to be expressed in meters and JX in
seconds of arc. If s is expressed in feet its logarithm must be reduced by 0.516 before using
in this table.
366
GEODETIC SURVEYING
TABLE III.— BAROMETRIC ELEVATIONS'
30
Containing H = 62737 log
B
B.
Inches.
11.0
11.1
.11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
12.0
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
13.0
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
14.0
H.
Feet.
27,336
27,090
26,846
26,604
26,364
26,126
25,890
25,656
25,424
25,194
24,966
24,740
24,516
24,294
24,073
23,854
23,637
23,421
23,207
22,995
22,785
22,576
22,368
22,162
21,958
21,757
21,557
21,358
21,160
20,962
20,765
Dif. for
.01.
Feet.
24.6
24.4
24.2
24.0
23.8
23.6
23.4
23.2
23.0
22.8
22.6
22.4
22.2
22.1
21.9
21.7
21.6
21.4
21.2
21.0
20.9
20.8
20.6
20.4
20.1
20.0
19.9
19.8
19.8
19.7
Inches.
14.0
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9
15.0
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
16.0
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
16.9
17.0
H.
Feet.
20,765
20,570
20,377
20,186
19,997
19,809
19,623
19,437
19,252
19,068
18,886
18,705
18,525
18,346
18,168
17,992
17,817
17,643
17,470
17,298
17,127
16,958
16,789
16,621
16,454
16,288
16,124
15,961
15,798
15,636
15,476
Dif. for
.01.
Feet.
19.5
19.3
19.1
18.9
18.8
18.6
18.6
18.5
18.4
18.2
18.1
18.0
17.9
17.8
17.6
17.5
17.4
17.3
17.2
17.1
16.9
16.9
16.8
16.7
16.6
16.4
16.3
16.3
16.2
16.0
Inches.
17.0
17.1
17.2
17.3
17.4
17.5
17.6
H.
17.7
17.8
17.9
18.0
18.1
18.2
18.3
18.4
18.5
18.6
18.7
18.8
18.9
19.0
19.1
19.2
19.3
19.4
19.5
19.6
19.7
19.8
19.9
20.0
Feet.
15,476
15,316
15,157
14,999
14,842
14,686
14,531
14,377
14,223
14,070
13,918
13,767
13,617
13,468
13,319
13,172
13,025
12,879
12,733
12,689
12,445
12,302
12,160
12,018
11,877
11,737
11,598
11,459
11,321
11,184
11,047
Dif. for
.01.
Feet.
16.0
15.9
15.8
15.7
15.6
15.5
15.4
15.4
15.3
15.2
15.1
15.0
14.9
14.9
14.7
14.7
14.6
14.6
14.4
14.4
14.3
14.2
14.2
14.1
14.0
13.9
13.9
13.8
13.7
13.7
* From Appendix No. 10, Report for 1881, United States Coast and Geodetic Survey.
TABLES
367
TABLE III.— BAROMETRIC ELEVATIONS— (Con^mweo)
30
Containing ff = 62737 log
B
B.
H.
Dif. for
.01.
B.
H.
Dif. for
.01.
B.
¥■
Dif. for
.01.
Inches.
Feet.
Feet.
Inches.
Feet.
Feet.
Inches.
Feet.
Feet.
20.0
11,047
13.6
23.0
7,239
11.8
26.0
3,899
10.5
20.1
10,911
13.5
13.4
23.1
7,121
11.7
26.1
3,794
10.4
20.2
10,776
23.2
7,004
11.7
26.2
3,690
10.4
20.3
10,642
13.4
23.3
6,887
11.7
26.3
3,586
10.3
20 4
10,508
13.3
23.4
6,770
11.6
26.4
3,483
10.3
20.5
10,375
23.5
6,654
26.5
3,380
13.3
11.6
10.3
20.6
10,242
13.2
23.6
6,538
11.5
26.6
3,277
10.2
20.7
10,110
13.1
23.7
6,423
11.5
26.7
3,175
10.2
20.8
9,979
13.1
23.8
6,308
11.4
26.8
3,073
10.1
20.9
9,848
13.0
23.9
6,194
11.4
26.9
2,072
10.1
21.0
9,718
12.9
24.0
6,080
11.3
27.0
2,871
10.1
21.1
9,589
12.9
24.1
5,967
11.3
27.1
2,770
10.0
21.2
9,460
12.8
24.2
5,854
11.3
27.2
2,670
10.0
21.3
9,332
12.8
24.3
5,741
11.2
27.3
2,570
10.0
21.4
9,204
12.7
24.4
5,629
11.1
27.4
2,470
9.9
21.5
9,077
12.6
24.5
5,518
11.1
27.5
2,371
9.9
21.6
8,951
12,6
24.6
5,407
11.1
27.6
2,272
9.9
21.7
8,825
12.5
24.7
5,296
11.0
27.7
2,173
9.8
21.8
8,700
12.5
24.8
5,186
10.9
27.8
2,075
9.8
21.9
8,575
12.4
24.9
5,077
10.9
27.9
1,977
9.7
22.0
8,451
12.4
25.0
4,968
10.9
28.0
1,880
9.7
22.1
8,327
12.3
25.1
4,859
10.8
28.1
1,783
9.7
22.2
8,204
12.2
25.2
4,751
10.8
28.2
1,686
9.7
22.3
8,082
12.2
25.3
4,643
10.8
28.3
1,589
9.6
22.4
7,960
12.2
25.4
4,535
10.7
28.4
1,493
9.6
22.5
7,838
12.1
25.5
4,428
10.7
28.5
1,397
9.5
22.6
7,717
12.0
25.6
4,321
10.6
28.6
1,302
9.5
22.7
7,597
12.0
25.7
4,215
10.6
28.7
1,207
9.5
22.8
7,477
11.9
25.8
4,109
10.5
28.8
1,112
9.4
22.9
7,358
11.9
25.9
4,004
10.5
28.9
1,018
9.4
23.0
7,239
26.0
3,899
29.0
924
368
GEODETIC 8UEVEYING
TABLE III.— BAROMETRIC ELEVATIONS— Conimued
30
Containing H = 62737 log — .
B.
H.
Dif. for
.01.
B.
H.
Dif. for
.01.
B.
H.
Dif. for
.01,
Inches.
Feet.
Feet.
Inches.
Feet.
Feet
Inches.
Feet.
Feet.
29.0
29.1
924
830
9.4
9.4
9.3
9.3
9.2
9.2
9.2
29.7
29.8
274
182
9.2
9.1
9.1
9.1
9.0
9.0
9.0
30.4
30.5
361
451
9.0
8.9
8.9
8.8
8.8
8.8
29.2
29.3
736
643
29.9
30.0
91
00
30.6
30.7
540
629
29.4
550
30.1
 91
30.8
717
29.5
29.6
29.7
458
366
274
30.2
30.3
30.4
181
271
361
30.9
31.0
805
893
TABLE IV.— CORRECTION COEFFICIENTS TO BAROMETRIC
ELEVATIONS FOR TEMPERATURE (FAHRENHEIT) AND
HUMIDITY *
t+v
C
t+t'
C
t+l'
C
0°
0.1025
60°
0.0380
120°
+0.0262
5
0.0970
65
0.0326
125
+0.0315
10
0.0915
70
0.0273
130
+0.0368
15
0.0860
75
0.0220
135
+0.0420
20
0.0806
80
0.0166
140
+0.0472
25
0.0752
85
0.0112
145
+0.0524
30
0.0698
90
0.0058
150
+0.0575
35
0.0645
95
0.0004
155
+0.0626
40
0.0592
100
+0.0049
160
+0.0677
45
0.0539
105
+0.0102
165
+0.0728
50
0.0486
110
+0.0156
170
+0.0779
55
0.0433
115
+0.0209
175
+0.0829
60
0.0380
120
+0.0262
180
+0.0879
* Based on Tables I and IV, Appendix No. 10, Report for 1881, United States Coast
and Geodetic Survey.
TABLES
369
TABLE v.— LOGARITHMS OF RADIUS OF CURVATURE
(In U. S. Legal Meters)
Latitude.
Azimuth.
24°
26°
28°
30°
32°
0°
180°
Meridian
6.802484
6.802602
6 802726
6.802857
6.802993
5
175
185°
355°
2503
2620
2744
2874
3009
10
170
190
350
2558
2674
2796
2924
3057
15
165
195
345
2649
2761
2880
3005
3135
20
160
200
340
2771
2880
2995
3116
3241
30
150
210
330
3098
3197
3301
3410
3523
40
140
220
320
3501
3585
3676
3771
3869
50
130
230
310
6.803928
6.803999
6.804075
6.804155
6.804238
60
120
240
300
4330
4389
4451
4517
4585
70
110
250
290
4658
4707
4758
4812
4868
75
105
255
285
4781
4827
4874
4923
4974
80
100
260
280
4872
4914
4958
5004
5052
85
95
265
275
4928
4968
5011
5054
5101
90
Prime Vert.
270
4947
4986
5028
5071
5117
34°
36°
38°
40°
42°
0°
180°
Meridian
6.803134
6.803279
6.803427
6.803578
6.803731
5
175
185°
355°
3150
3294
3441
3591
3744
10
170
190
350
3195
3337
3483
3631
3780
15
165
195
345
3270
3409
3551
3695
3840
20
160
200
340
3371
3505
3642
3781
3922
30
150
210
330
3641
3762
3885
4011
4138
40
140
220
320
3972
4077
4184
4294
4405
50
130
230
310
6.804324
6.804412
6.804503
6.804595
6.804688
60
120
240
300
4655
4728
4802
4878
4954
70
110
250
290
4926
4985
5046
5109
5171
75
105
255
285
5027
5081
5138
5195
5253
80
100
260
280
5102
5153
5206
5259
5313
85
95
265
275
5148
5197
5247
5299
5350
90
Frime Vert,
270
5164
5212
5261
5312
5363
44°
46°
48°
50°
52°
0°
180°
Meridian
6.803885
6.804040
6:804194
6.804347
6.804498
5
175
185°
355°
3897
4050
4204
4356
4506
10
170
190
350
3931
4082
4233
4383
4531
15
165
195
345
3987
4135
4282
4428
4573
20
160
200
340
4064
4206
4348
4489
4629
30
150
210
330
4267
4396
4524
4652
4778
40
140
220
320
4516
4628
4740
4851
4960
50
130
230
310
6.804782
6.804876
6.804970
6.805063
6,805155
60
120
240
300
5030
6109
5186
5262
5338
70
110
250
290
5234
5298
5362
5425
5487
75
105
255
285
5312
5369
5428
5486
■ 5543
80
100
260
280
5368
5422
5477
5531
6584
85
95
265
275
5402
5455
5507
5559
5610
90
Prime Vert.
270
5414
5465
5517
5568
5618
370
GEODETIC SUEVEYING
TABLE VI.— LOGARITHMS OP RADIUS OF CURVATURE
(In feet)
Azimuth.
Latitude.
28°
30°
32°
34°
36°
0°
180°
Meridian
7.318711
7.318841
7.318978
7.319118
7.319263
5
175
185°
355°
8728
8858
8993
9134
9278
10
170
190
350
8780
8908
9041
9179
9321
IS
165
195
345
8864
8989
9119
9254
9393
20
160
200
340
8979
9100
9225
9355
9489
30
150
210
330
9285
9394
9507
9625
9746
40
140
220
320
9660
9755
9853
9956
320061
50
130
230
310
7.320059
7.320139
7.320222
7.320308
7.320396
60
120
240
300
0435
0501
0569
0639
0712
70
110
250
290
0742
0796
0852
0910
0969
75
105
255
285
0858
0907
0958
1011
1065
80
100
260
280
0942
0988
1036
1086
1137
85
95
265
275
0995
1038
1085
1132
1181
90
Prime Vert.
270
1012
1055
1101
1148
1196
38°
40°
42°
44°
46°
0°
180°
Meridian
7.319412
7.319562
7.319715
7.319869
7.320024
5
175
185°
355°
9425
9575
9728
9881
0034
10
170
190
350
9467
9615
9764
9915
0066
15
165
195
345
9535
9679
9824
9971
0119
20
160
200
340
9626
9765
9906
320048
0190
30
150
210
330
9869
9995
320122
0251
0380
40
140
220
320
320168
320278
0389
0500
0612
50
130
230
310
7.320487
7.320579
7.320672
7.320766
7.320860
60
120
240
300
0786
0862
0938
1014
1093
70
110
250
290
1030
1093
1155
1218
1282
75
105
255
285
1122
1179
1237
1296
1353
80
100
260
280
1190
1243
1297
1352
1406
85
95
265
275
1231
1283
1334
1386
1439
90
Prime Vert.
270
1246
1296
1347
1398
1449
TABLE VII.— CORRECTIONS FOR CURVATURE AND REFRACTION
IN PRECISE SPIRIT LEVELING
Correction
Correction
Correction
Distance.
to Rod
Distance.
to Rod
Distance.
to Rod
Reading.
Reading,
Reading.
Meters.
mm.
Meters.
mm.
Meters.
mm.
Oto 27
0.0
100
0.68
200
2.73
28 to 47
0.1
no
0.83
210
3.01
48 to 60
0.2
120
0.98
220
3.31
61 to 72
0.3
130
1.15
230
3.61
73 to 81
0.4
140
1.34
240
3.94
82 to 90
0.5
150
1.54
250
4.27
91 to 98
0.6
160
1.75
260
4.62
99 to 105
0.7
170
1.97
270
4.98
106 to 112
0.8
180
2.21
280
6.36
113 tolls
0.9
190
2,47
290
5.75 
TABLES
TABLE VIIL— MEAN ANGULAR REFRACTION
371
Apparent
Altitude.
Refraction.
Apparent
Altitude.
Refraction.
Apparent
Altitude,
Kef r action.
Apparent
Zenith
Distance.
o /
/
ft
o
/ //
o
/
It
o
00
34
54.1
10
5 16.2
50
48,4
40
10
32
49.2
11
4 48.6
51
46.7
39
20
30
52.3
12
4 25.0
52
45.1
38
30
29
03.5
13
4 04.9
53
43.5
37
40
27
22.7
14
3 47.4
54
41.9
36
50
25
49.8
16
3 32.1
55
40.4
35
1 00
24
24.6
16
3 18.6
56
38.9
34
10
23
06.7
17
3 06.6
57
.
37.5
33
20
21
55.6
18
2 65.8
58
36.1
32
30
20
50.9
19
2 46.1
59
34.7
31
40
19
51.9
60
18
58.0
20
2 37.3
60
33.3
30
21
2 29.3
61
32.0
29
2 00
18
08.6
22
2 21.9
62
30.7
28
10
17
23.0
23
2 16.2
63
29.4
27
20
16
40.7
24
2 08.9
64
28.2
26
30
16
00.9
40
15
23.4
26
2 03.2
65
26.9
26
50
14
47.8
26
1 67.8
66
25.7
24
27
1 52.8
67
24.5
23
3 00
14
.14.6
28
1 48.2
68
23.3
22
10
13
43.7
29
1 43.8
69
22.2
21
20
13
15.0
30
12
48.3
30
1 39.7
70
21.0
20
40
12
23.7
31
1 35.8
71
19.9
19
50
12
00.7
32
1 32,1
72
18.8
18
33
1 28.7
73
17.7
17
4 00
11
38.9
34
1 26.4
74
16.6
16
10
11
18.3
20
10
58.6
35
1 22.3
75
15.5
16
30
10
39.6
36
1 19,3
76
14.5
14
40
10
21.2
37
1 16.5
77
13.4
13
50
10
03.3
38
1 13,8
78
12,3
12
39
1 11,2
79
11,2
11
5 00
9
46.5
30
9
01.9
40
1 08.7
80
10.2
10
41
1 06,3
81
09.1
9
6 00
8
23,3
42
1 04.0
82
08.1
8
30
7
49.5
43
1 01.8
83
07.1
7
44
59.7
84
06.1
6
7 00
7
19.7
30
6
63.3
45
67.7
85
05.1
5
46
55.7
86
04,1
4
8 00
6
29.6
47
53,8
87
03,0
3
30
6
08.4
48
51,9
88
02.0
2
49
50.2
89
01.0
1
9 00
6
49.3
30
6
32.0
60
48,4
90
00,0
372
GEODETIC SUEVEYING
TABLE IX.— ELEMENTS OF MAP PROJECTIONS
Lat.
Logarith
ns (U. S. Legal Meters).
1° in Meters.
Logarithm
(1e'sinZ^).
(10)
B
N
r
Latitude.
(<^30' to
.#. + 30')
Longitude.
(On Par. of
Latitude.)
20°
22
24
26
28
6.8022696
3727
4835
6015
7262
6.8048752
9096
9465
9859
6.8050274
6.7778610
.7720755
.7656767
.7586461
.7509623
110700
726
754
785
816
104650
103265
101755
100121
98365
9.9996560
5873
5134
4347
3516
30
32
34
36
38
6.8028569
9930
6.8031339
2788
4271
6.8050710
1164
1633
2116
2611
6.7426016
.7335369
.7237375
.7131692
. 7017932
110850
884
920
957
995
96489
94496
92388
90167
87836
9.9992645
.1738
0798
9.9989832
8843
40
42
44
46
48
6.8035781
7309
8849
6.8040393
1934
6.8053114
3623
4136
4651
5165
6.6895654
.6764358
.6623477
.6472364
.6310274
111034
073
112
152
191
85397
82854
80209
77466
74629
9.9987837
6818
5792
4762
3735
50
52
54
56
58
6.8043463
4975
6460
7913
9326
6.8055675
6178
6674
7158
7629
6.6136350
.5949598
.5748861
. 5532775
. 5299726
111231
269
307
345
381
71699
68681
65579
62396
59136
9.9982715
1708
0717
9.9979749
8807
60
6.8050691
6.8058084
6 . 5047784
111416
55803
9.9977897
Lat.
Element
of
Coordinates of Developed Arcs. 1
^
V
Cone.
for 1° of Long.
for 71° of Long.
for X? of Long.
for 71°.
Miles.
Miles.
Meters.
Value for (1°) X
Miles.
Meters.
(1°) X
20°
10893
65.03
104649
n 003(0.19771°)
0.1941
312.3
7l2
22
9814
64.17
103264
n.cos (0.216«°)
. 2098
337.6
71!
24
8907
63.23
101754
nC03 (0.235n°)
0.2244
361.2
71'
26
8131
62.21
100120
mcos (0,253n°)
0,2380
383.0
71'
28
7459
61.12
98364
ncos(0.271«°)
0.2504
403.0
712
30
6870
59.95
96488
n 008(0.28871°)
0.2616
421,0
712
32
6349
58.72
94495
noos (0.30571°)
0.2715
437,0
71!
34
5882
57.41
92386
71 cos (0.322n°)
0.2801
450.8
71!
36
5461
56.03
90165
71 cos (0.33971°)
0.2874
462,5
712
38
5079
54.58
87834
71 cos (0.35571°)
0.2932
471,9
712
40
4730
53.06
85395
7icos(0.371?i°)
0.2976
479.0
71'
42
4408
51.48
82852
71 cos (0.38671°)
0.3006
483.8
7l2
44
4111
49.84
80207
7ioos(0.400n°)
0.3021
486.2
712
46
3834
48.13
77464
71 cos (0.41471°)
. 3022
486.3
712
48
3575
46.37
74627
71 cos (0.42871°)
0.3007
484.0
712
50
3332
44.55
71697
71 cos (0. 44171°)
. 2978
479.3
7l2
52
3103
42.68
68679
ncos (0. 45471°)
0.2935
472.3
712
54
2886
40.75
65577
71  cos (0. 46671°)
0.2877
463.0
712
56
2679
38,77
62394
71 cos (0.47871°)
0.2805
451.4
712
58
2483
36.74
59134
71. cos (0.48971°)
0.2719
437.6
712
60
2294
34.67
55801
71 cos (0.49971°)
0.2620
421.7
712
TABLES
373
TABLE X.— CONSTANTS AND THEIR LOGARITHMS
General Constants.
Number.
Logarithm.
7r
3.141592654
0.318309886
9.869604401
0.101321184
1.772453851
0.564189584
57.29577951
3437,746771
206264.8062
0.017453293
0.017452406
0.000290888
0.000290888
0.000004848
0.000004848
2.718281828
0.434294482
0.434294482
2.302585093
3,2808693,.
3.280833333
0.621369949
1609.347219
0.4769363..
0.6744897..
0.4971498727
9,5028501273
0.9942997454
9.0057002546
0.2485749363
9.7514250637
1.7581226324
3.5362738828
5.3144251332
8,2418773676
8.2418553284
6.4637261172
6.4637261109
4.6855748668
4.6855748668
0.4342944819
9.6377843113
9.6377843113
0.3622156887
0,5159889297
0.5159841687
9.7933502462
3.2066497538
9.6784604...
9.8289754...
10
10
10
10
10
10
10
10
10
10
10
10
10
10
1
7^
TT^
1
k2
Vtt
1
sfk
Degrees in a radian
Minutes in a radian
Seconds in a radian
Arcl"
Sin 1°
Sin 1'
Arc 1"
Modulus of common logarithms (M)
Natural log x rcommon log x
1 U. S. legal meter = 3.2808333 + ft
1 kilometer = fiveeighths mile, nearly . . .
Geodetic Constants.
(Clarke's 1866 Spheroid.)
Logarithms.
U. S. Legal Meters.
Feet. 1
Semimajor axis = o . . .
6.8047033
6.8032285
9.9985252
6.8039665
7.5302093
8.9152513
7.8305026
9.9970504
6.8017537
6.8061781
10
10
10
10
10
7.3206875
7.3192127
9.9985252
7.3199507
7.5302093
8.9152513
7.8305026
9.9970504
7.3177379
7.3221623
10
10
10
10
10
Semiminor axis — 6~o ^\ — e
Ratio of axes293.98r 294,98
Mean radius
Ellipticity = = e
/o2 62
L^=l_e2
a2
^!=„(l_e,)
o2 a
6 Vi  e2
BIBLIOGRAPHY
REFERENCES ON GEODETIC SURVEYING
Adjustment of Observations, Wright and Hayford. D. Van Nostrand &
Co., New York, 1904.
Elements of Geodesy, Gore. John Wiley & Sons, New York, 1893. Gillespie's
Higher Surveying, Staley. D. Appleton & Co., New York, 1897.
Johnson's Theory and Practice of Surveying, Smith. John Wiley & Sons,
New York, 1910.
Manual of Spherical and Practical Astronomy, Chauvenet. J. B. Lippin
cott & Co., Philadelphia, 1885.
Practica' Astronomy as Apphed to Geodesy and Navigation, DooUttle.
John Wiley & Sons, New York, 1893.
Precise Surveying and Geodesy, Merriman. John Wiley & Sons, New
York, 1899.
Principles and Practice of Surveying, Breed and Hosmer. John Wiley &
Sons, New York, 1906.
Text Book of Field Astronomy for Engineers, Comstock. John Wiley &
Sons, New York, 1902.
Text Book of Geodetic Astronomy, Hayford. John Wiley & Sons, New York,
1898.
Text Book on Geodesy and Least Squares, Crandall. John Wiley & Sons,
New York, 1907.
Geodesic Night Signals, Appendix No. 8, Report for 1880, U. S. Coast and
Geodetic Survey.
Field Work of the Triangulation, Appendix No. 9, Report for 1882, U. S.
Coast and Geodetic Survey.
Observing Tripods and Scaffolds, Appendix No. 10, Report for 1882, U. S.
Coast and Geodetic Survey.
Geodetic Reconnaissance, Appendix No. 10, Report for 1885, U. S. Coast
and Geodetic Survey.
Relation of the Yard to the Meter, Appendix No. 16, Report for 1890, U. S.
Coast and Geodetic Survey.
Fundamental Standards of Length and Mass, Appendix No. 6, Report for
1893, U. S. Coast and Geodetic Survey.
Perfected Form of Base Apparatus, Appendix No. 17, Report for 1880, U. S.
Coast and Geodetic Survey.
374
BIBLIOGEAPHY 375
Description of a Compensating Base Apparatus, Appendix No. 7, Report
for 1882, U. S. Coast and Geodetic Survey.
The Eimbeck Duplex Basebar, Appendix No. 11, Report for 1897, U. S.
Coast and Geodetic Survey.
Measurement of Base Lines (Jaderin Method) with Steel Tapes and with
Steel and Brass Wires, Appendix No. 5, Report for 1893, U. S. Coast
and Geodetic Survey.
Measurement of Base Lines with Steel and Invar Tapes, Appendix No. 4,
Report for 1907, U. S. Coast and Geodetic Survey.
Run of the Micrometer, Appendix No. 8, Report for 1884, U. S. Coast and
Geodetic Survey.
Synthetic Adjustment of Triangulation Systems, Appendix No. 12, Report
for 1892, U. S. Coast and Geodetic Siirvey.
Formulas and Tables for the Computation of Geodetic Positions, Appendix
No. 9, Report for 1894, and Appendix No. 4, Report for 1901, U. S.
Coast and Geodetic Survey.
Barometric Hypsometry, Appendix No. 10, Report for 1881, U. S. Coast
and Geodetic Survey.
Transcontinental Line of Levehng in the United States, Appendix No. 11,
Report for 1882, IT. S. Coast and Geodetic Survey.
SeHregistering Tide Gauges, Appendix No. 7, Report for 1897, U. S. Coast
and Geodetic Survey.
Precise Leveling in the United States, Appendix No. 8, Report for 1899, and
Appendix No. 3, Report for 1903, U. S. Coast and Geodetic Survey.
Variations in Latitude, Appendix No. 13, Report for 1891, Appendix No. 1,
Report for 1892, Appendix No. 2, Report for 1892, and Appendix No.
11, Report for 1893, U. S. Coast and Geodetic Survey.
Tables of Azimuth and Apparent Altitude of Polaris, Appendix No. 10,
Report for 1895, U. S. Coast and Geodetic Survey.
Determination of Time, Latitude, Longitude, and Azimuth, Appendix No. 7,
Report for 1898, U. S. Coast and Geodetic Survey.
A Treatise on Projections, U. S. Coast and Geodetic Survey, 1882.
Tables for the Polyconic Projection of Maps (Clarke's 1866 Spheroid),
Appendix No. 6, Report for 1884, U. S. Coast and Geodetic Survey.
Geographical Tables and Formulas, U. S. Geological Survey, 1908.
Bibliography of Geodesy (Gore), Appendix No. 8, Report for 1902, U. S.
Coast and Geodetic Survey.
REFERENCES ON METHOD OF LEAST SQUARES.
Manual of Spherical and Practical Astronomy, Chauvenet. J. B. Lippincott
Co., PhOadelphia, 1885.
Approximate Determination of Probable Error, Appendix No. 13, Report
for 1890, U. S. Coast and Geodetic Survey.
Theory of Errors and Method of Least Squares, Johnson. John Wiley &
Sons, New York, 1893.
Practical Astronomy as Applied to Geodesy and Navigation, Doolittle.
John Wiley & Sons, New York, 1893.
376 BIBLIOGEAPHY
Precise Surveying and Geodesy, Merriman. John Wiley & Sons, New
York, 1899.
Adjustment of Observations, Wright and Hayford. D. Van Nostrand &
Co., New York, 1904.'
Text Book on Geodesy and Least Squares, Crandall. John Wiley & Sons,
New York, 1907.
INDEX
PAGE
Aberration of light (diurnal) 213
Absolute length, correction for 36
Absolute locations 4
Accidental errors:
laws of . 252
nature of 247
theory of 252265
Accuracy attainable in
angle measurements 78
barometric leveling 129
baseline measurement 45
closing triangles 102
precise spirit leveling 161
trigonometric leveling 139
Adjustment of
angle measiuements 81, 100, 312—332
baseline measurements. 333
level work 160, 344359
observations 3, 241359
quadrilaterals 90100, 327
triangles 89, 322—326
Adjustments of
Coast Survey precise level 155156
direction instrument 65
European type of precise level 146152
repeating instrument 59
Alignment corrections:
• horizontal 40
vertical '^2
Alignment curve 64
Altazimuth iastnmient 48, 51
Altitude 167
American Ephemeris 164
Aneroid barometer 126, 127
377
378 INDEX
FAOB
Angles:
accuracy of measurements 78
adjustment of 81, 100, 312332
eccentric 75
exterior and interior , 53
instruments for measuring 47, 52, 60
measurement of 4780
Apparent time 165
Arcs, elliptic 108
Arithmetic mean 244
Associations, geodetic 1
Astronomical determinations 163226
See also Azimuth, Latitude, Longitude, and Time.
Azimuth 4, 109, 167, 203
astronomical 204
geodetic 204
lines, planes and sections Ill
marlfs 204
periodic changes in 226
Azimuthal angles 109, 117
Azimuth determinations 203226
approximate 214
at sea 225
by meridian altitudes of sun or stars 205
by observations on circumpolar stars 207225
direction method 215
fimdamental formulas 208
micrometric method 221
repeating method 218
Back azimuth 109, 113, 122
errors 122
Barometers, aneroid and mercurial 126, 127
Barometric leveling 125, 126130
Basebars 2429
compensating 26
Eimbeck duplex 26
general features of 25
standardizing 33
thermometric 26
tripods for 27
BaseUne measurements 2446, 333343
accuracy of 45
adjustment of 333
check bases 5
corrections required 24, 3544
duplicate hnes 334
INDEX 379
PAGB
Baseline measurements — {continued)
gaps, computing length of 44
general law of probable error 336
law of relative weight 337
probable error of . . . , 65
probable error of lines of unit length . ". 338, 339
sectional lines 335
standardizing bars and tapes 33
with basebars 24r29
with steel and brass wires 32
with steel and invar tapes 3032
uncertainty of 46, 342
Bessel's solution of geodetic problem 118
Bessel's spheroid 106
Bibhography 374
Board signals 20
Bonne's map projection 238
Celestial sphere 166
Chance, laws of 248
Changes, periodic:
in azimuth 226
in latitude 1.96
in longitude 203
Check bases 5
Chronograph .' 184
Circumpolar stars 190, 207
Clarke's spheroid 106
Clarke's solution of geodetic problem:
direct 116
inverse 118
Closed level circuits 160, 357
Closing the horizon 53, 313, 315
Coast and Geodetic Survey, United States 1
papers of 1
precise level 153
Coefficient of refraction 138
Cofunctions:
altitude 167
declination , 167
latitude 167
Comparator 34
Compensating basebars 26
Computation of geodetic positions 103124
Bessel's solution 118
Clarke's solution 116
Helmert's solution 118
380 INDEX
PAOB
Computations of geodetic positions — {continued)
inverse problem • 118
Puissant's solution 113
Computed quantities:
most probable values of 296
probable errors of ' 306311
Conditional equations 284
Conditioned quantities:
definition of 242, 284
most probable values of 284295
probable errors of 304
Convergence of meridians 88, 111
Corrections in baseline work 24, 3544
Correlative equations 290
Crosssection of tapes 38
Culmination, meaning of 190
Curvature and refraction (in elevation) 12
Declination 167
Degree, length of:
meridian 228
parallel of latitude 228
Dependent equations 284
Dependent quantities:
definition of 242, 284
most probable values of 284295
probable errors of 304
Deviation of plumb line 124
Dip of horizon 184
Direction instrument 47, 50, 60
adjustments of 71
Direct observations 243
Distances, polar and zenith 167
Diiunal aberration 213
Duplex basebars 26
DupUcate base lines 334
Duplicate level lines 160, 346
Earth, figure of:
general figure 104
practical figure 106
precise figure 105
Eccentric signals 20, 78
stations : 75
Eimbeck duplex basebar 26
Elevation of stations 62
Ellipsoid, definition of 105
INDEX 381
FAOE
EUiptio arcs 108
Elongation, definition of 208
Ephemeris, American ■ 164
Equation of time 165
Equations :
conditional 284
correlative 290
dependent 284
normal 273, 275
observation 271
probability > 257
reduced observation 281
Errors:
classification of 245, 247
facility of 255
in precise leveling 143
laws of 252
probability of 256
theory of : 252265
types of 254
European precise level 141, 145
adjustments of 146152
Exterior angles 53
Figure adjustment 81, 87, 100, 312, 321332
Figure of earth:
analytical considerations 110
constants of 106
general figure 104
geometrical considerations 106
practical figure 106
precise figure 105
Filar micrometer:
description of 66
reading the micrometer , 67
run of the micrometer 68
Flattening of the Earth's poles 104, 105
Foot pins and plates 158
Gaps in base lines 44
Geodesic liae 109
Geodesy:
definition of 1
history of 1
scope of 2
Geodetic associations 1
Geodetic leveling 125162
382 INDEX
PAOE
Geodetic map drawing 227240
Geodetic positions, computation of 103124
Geodetic quadrilateral 7, 90, 327
Geodetic surveyiug 1240
Geodetic work in the United States , 1
Geoid, definition of 106
Geometric mean 244
HarrebowTalcott latitude method 193
Heat radiation 47
Height of stations 17
Heliotropes 21
Helmert's solution of geodetic problem 118
History of plane and geodetic surveyiag 1
Horizontal alignment 40
Hour angle .• 164, 167
Independent quantities:
definition of 241
most probable values of 266283
probable errors of 300304
Indirect observations 243
Instruments, geodetic; see Angles, Astronomical determinations. Base
line measurements and geodetic leveUng.
Interior angles 53
Intermediate points in leveling 160, 217
International Geodetic Association 1
Intervisibility of stations 11, 14
Invar tapes 32
Inverse geodetic problem 118
Jaderin baseUne methods:
with tapes 31
with wires 32
Latitude 109, 167, 186
astronomical 186
geocentric 187
geodetic 186
locating a parallel of 120
periodic changes in 196
Latitude determinations 188196
at sea 196
by circumpolar culminations 190
by HarrebowTalcott method 193
by meridian altitudes of sun 188
by prime vertical transits 192
by zenith telescope 193
INDEX 383
PAGE
Law of
coefficients in correlative equations 294
coefficients in normal equations 280
facility of error 257
Laws of
chance 248250
errors 252
weights 82
Least squares, method of 241359
Lengths of bars and tapes 24, 33
Leveling:
barometric 125, 126130
geodetic 125162
precise spirit 125, 139162
trigonometric 125, 130139
Level work:
adjustments 160, 344359
branch hnes, circuits and nets 359
closed circuits 160, 357
duphcate hnes 160, 346
general law of probable error 347
intermediate points : 160, 355
law of relative weight 348
level nets 161, 352
multiple hnes 160, 350
probable error of lines of unit length 348, 349
sectional hnes 347
simultaneous lines 160
Light, diurnal aberration of 213
L. M. Z. problem 103
Locating a parallel of latitude 120
Locations, absolute and relative 4
Longitude 109, 197
astronomical 197
geodetic 197
periodic changes in *. 203
Longitude determinations 197203
at sea 203
by lunar observations 198
lunar culminations 199
lunar distances 199
lunar occultations 199
by special methods 198
flash signals 198
special phenomena 198
by telegraph 200
arbitrary signals 202
384 INDEX
PAGE
by telegraph — (continued)
standard time signals 201
star signals 201
by transportation of chronometers 199
Loxodrome 233
Map projections 227240
conical 234
Bonne's projection 238
Mercator's conic '. 236
simple conic 235
cylindrical ' .'229
Mercator's cylindrical 231
rectangular cylindrical 231
simple cyUndrioal i 229
polyconic 240
rectangiilar polyconic 241
simple polyconic 240
trapezoidal 234
Mean absolute error .....' 305
Mean error 305
Mean of errors 305
Mean radius of the earth 44
Mean sea level 43, 125
Mean solar time 165
Measures of precision 262, 304
Mercator's projections:
conic 236
cylindrical 231
Mercurial barometer 126, 127
Meridian 167
lengths 228
line, plane, and section 167
Meridians, convergence of 88, 111
Method of least squares 241359
Micrometer:
filar 66
microscope 65
reading of 67
run of 68
Mistakes 247
INDEX 385
FAQB
Most probable values of — {continued)
independent quantities 266283
observed quantities 242, 266, 295
Multiple level lines 169, 360
Nadir 167
Nautical Almanac 164
Night signals 23
Normal ■ 110
Normal equations 273, 276
law of coefficients 280
Normal tension 40
Observation equations:
definition of 271
reduced 281
reduction to unit weight 278
Observations:
adjustment of 3, 241359
classification of 243
Observed quantities:
most probable values of 266295
probable errors of 297305
Observed values, definition of 242
OvalcSd, definition of 105
Papers of U. S. Coast and Geodetic Survey 1
Parallax (in altitude) 167, 171
Parallel of latitude, location of 120
Parallels, length of one degree 228
Phase 20
Phaseless targets 20
Plane surveying, history of 1
Plumbline deviation 124
Polar distance 167
Pole signals 20
Precise spirit leveling 125, 139162
accuracy attainable 161
adjustment of results ; .' 160, 344359
n^Oof ail^TQTr TllOnJoO loTIol 1 d.9 IRS
386 INDEX
PAGE
Precise spirit leveling — {continued)
instruments used 139, 145, 153
methods 143, 145
rods and turning points 158
sources of error 143
Primary triangles and systems 9
Prime vertical 110, 167
Primevertical transits 192
Probability:
equation of 257, 260
laws of chance 248
Probable error:
general value of 29*9
meaning of 297
Probable errors of
angle measurements 79
baseUne measurements 46
computed quantities 306311
conditioned quantities 304
dependent quantities 304
independent quantities 300304
observed quantities 297305
Projection of maps 227240
See Map projections for list of types.
Puissant' s solution of geodetic problem:
direct 113
inverse 118
PuU, with tapes and wires 24, 30, 38
Quadratic mean 244
Quadrilateral, geodetic 7, 90, 327
algebraic adjustment of 90102
approximate 92
definition of rigorous 96
least square adjustment of 327
Quantities:
classification of 241
most probable values of 266296
computed quantities 296
observed quantities 266296
probable errors of 297311
computed quantities 306311
observed quantities 297305
Radiation, heat 47
Reading micrometers 67
Reconnoissance 10
INDEX 387
PAGE
Reduced observation equations 281
Reduction to center 75
Reduction to mean sea level 43
Refraction:
angular 167
coefficient of ■ 138
in elevation 12
Relative locations 4
Repeating instruments 47, 49, 62
adjustments of 59
Residual errors 245
Residuals 245
Rhumb line 233
Right ascension 167
Run of micrometer 68
Sag 24, 30, 39
Secondary triangles and systems 9
Sectional lines:
base hues 335
level lines 347
Sidereal time 165, 168
Signals at stations 18
board 20
eccentric 20, 78
hehotrope 21
night 23
phaseless 20
pole 20
Simultaneous level hnes 160
Single angle adjustment 312
Solar time 165
Spherical excess 88, 89, 90
Spheroid:
Bessel's , 106
Clarke's 106
definition of 105
Spirit leveling, see Precise spirit leveling.
Standardizing bars and tapes 33
Standard time 165
Station adjustment 81, 84, 312, 313319
Stations:
elevation of 14, 17
height ,of 17
intervisibility of 11, 14
marks 17
selection of 10
388 INDEX
PAGE
Stations — {continued)
signals and targets 18
towers 17, 18
triangulation 5
Steel tapes 24, 30, 32
corrections required in tape measurements 24, 3339
standardizing 33
Steel and brass wires 32
Systematic errors 247
Tables 361373
Tangents 110, 120
Targets 18
Telegraphic determination of longitude 200
Telescope, zenith 193
Temperature corrections in baseline work 24, 31, 36
Tension, tapes and wires 40
Tertiary triangles and systems 9
TheodoHte 48
Theory of errors ' 252265
comparison of theory and experience 264
Theory of weights 81, 243
Thermometric basebars 26
Tide gauges:
automatic 125
staff 126
Time 164
conversion of 165, 169, 170
general principles 164
varieties of 165
Time determinations 164186
at sea 184
by equal altitudes of sun V . 176
by siagle altitudes of sun 171
by sun and star transits 181
choice of methods 184
Towers, station and signal 17, 18, 47
Transit, astronomical 183, 185
Triangles:
accuracy in closing 102
adjustment of 89, 322326
classification of 9
computation of 102
Triangulation:
adjustments and computations 81102, 312332
general scheme 4
principles of , 423
INDEX 389
FAOB
Triangulation — (continued)
stations 5, 10
systems 59
Trigonometrical leveling 125, 130139
accuracy attainable 139
observations at one station 133
reciprocal observations 136
searhorizon method 131
Tripods for
anglemeasuring instruments 18
basebars 27
leveling instruments 143
True errors 245
True values 242
Turning points 158
Uncertainty of baseline measurements 46, 342
United States Coast and Geodetic Survey 1
papers of 1
precise level 153
Values, classification of 242, 244
Variations, periodic:
in azimuth 226
in latitude 196
in longitude 203
Vertical alignment 42
Weight:
laws of 82
theory of 81, 243
Wires, steel and brass 32
Zenith 167
Zenith distance 167
Zenith telescope 193