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OCR for page 221
CIIAPTI2R XV
THE DETERMINATION OF GEOGRAPHIC POSITIONS
C. V. HODGSON (b. 1880. d. 1929)
Formerly in the U. S. Coast and Geodetic Survey.
In the early days of man's existence it was found necessary to know
the distance and direction between widely separated points and when
he became more civilized and especially when he began agricultural
pursuits the location of boundary lines between parcels of land became
necessary or, at least, desirable. In the highly organized civilization
of today an exact knowledge of the location of places with respect to
each other is essential.
Reference system. For the past few centuries the generally accepted
method of designating the location of points on the earth's surface is
by latitudes and longitudes. The latitude of a point is its angular
distance north or south of the Equator. The longitude of a point is the
angular distance measured alone, the Equator between the meridian
through the point and the meridian through some reference station.
The reference station now almost universally used is the observatory at
Greenwich, England. The latitude and longitude of a point on the earth's
surface is called its geographic position.
Since the demands of the scientific man and the engineer for the exact
locations of points and the determinations of the shape and size of the
earth are very exacting, methods have been evolved to determine latitudes,
longitudes, directions, and distances, with extreme accuracy. Either of
two general methods may be emplo~ecl., the astronomical or the geodetic
and each will be briefly clescribed.
Astronorn~cal determ~rrat~or~ of qeo~raph~c pos1,t~ons. Since the hea.v-
enly bodies have been observed at astronomical observatories over long,
periods of time and their relative positions and motions determined and
the results catalogued, observations upon these celestial objects can be
used in the determination of the latitude and longitude of a point on
the earth's surface.
In ordinary surveying operations, on exploratory surveys, or when
navigating at sea, observations for latitude and longitude may be made
with small instruments on either the sun or stars. NIany methods are
available, dependin,, upon the kind of instrument used, the accuracy
desired and the conditions under which the observations are made. On
what is called geodetic astronomy, however, where the results are used
221
OCR for page 222
222
FI GUR E OF THE EAR TH
for the determination of the figure of the earth, -i'or the location of state
and national boundaries, and to furnish data used in the adjustment of
triangulation, special instruments are employed and all observations are
made upon stars. awhile the accuracy required necessarily limits the
choice of methods which may be employed in geodetic astronomy, the
principles involved are in the main the same as for lower grade work.
~ description will be given only of' those methods and instruments ordi-
narily used in the more precise determinations. of astronomical geo-
~,raphie positions.
Latitude. If one reverts to some definitions of elementary astronomy
and recalls that the extension of the plane of the earth's equator to the
celestial sphere traces the celestial equator, and the declinations of
. ~ . . , ~ .
the heavenly cockles are reckoned north and south from the celestial
equator in the same manner as latitudes on the earth, then it is clearly
seen that the declination of the zenith of a point on the earth is equal
to the latitude of the point. If now we measure the angle between the
zenith and some star as it crosses the meridian, and know the declina-
t.ion of the star, a simple computation gives the latitude of the place.
If the point is in the northern hemisphere and the star is south of the
zenith, the measured zenith distance plus the star's declination will
equal the latitude of the point. If the star is north of the zenith, or
the point is south of the Equator, the arithmetic relation is changed
but is equally simple. Of course there are instrumental corrections
to be applied and the observed angle must be corrected to what it would
have been had the observations been made at the center of the earth
instead of on its surface, but these are refinements rather than modi-
fications of the essential principle involved.
Until recently latitudes were determined by means of observations
made with an instrurrtent called the zenith telescope. (See Figure 1.)
But now an instrument is available called the broken telescope transit
~ see Figure 2 ~ with which latitudes can be observed. With either
instrument the well known Talcott method is employed. The charac-
teristic feature of this method is that instead of measuring the zenith
v
distance of a single star, the d~tterence In tile zenith distances of
two stars is measured micrometrically as they pass the meridian, the
two stars selected to -form a pair having approximately the same zenith
distance but being on opposite sides of the zenith. The mean of their
declinations, corrected for one half the difference of their zenith dis-
tances, would therefore be equal to the declination of the zenith or
the latitude o:: the place of' observation. This method practically elimi-
nates the errors due to the uncertainty of the coefficient of refraction
which affects the star's rays. Delicate levels are read when the pointing
OCR for page 223
DETERMINATION OF GEOGRAPHIC POSITIONS 223
F'IG. 1.-Zenith telescope. An instrument designed especially for determining
astronomical latitudes by the Talcott method.
15
OCR for page 224
221- FIGURE OF THE EARTII
is made on each star and corrections are applied for the inclination of
the telescope. From ~ ~ to 18 pairs of stars are easily observed i
3 to ~ hours and the resulting latitude has a probable error seldom
greater than 0".10. This accuracy is all that its necessary in geocletic
astronomy. An observing list of pairs of stars can be made out from a
star catalogue in a few hours. The instrument needs no other support
FIG. 2. Droller telescope transit. While the primary purpose of this instru-
ment is to determine time by observing the transit of stars across the meridian.
latitude can be determined by the Talcott method if the instrument is equipped
with the special sensitive levels which are shown in the picture.
than a tripod or pier made of wood or o:l' aluminum alloy SO joined
together as to form a rigid structure.
Observations are made with the zenith telescope for st.ucly of' the
variation of latitude where the accuracy of the observations is much
greater than that indicated above. As is well known the latitude of any
point on the earths surface varies from time to time, due to unknown
causes, and by an amount as great as 0".~. In order that. the law of the
variation might be discovered, observations have been made over a period
OCR for page 225
DETERMINATION OF GEOGRAPHIC POSITIONS 225
of years at several stations placed in approximately the same latitude.
The stations now in operation are at Ukiah, California; )¢i%usawa, Japan;
and Carloforte, Italy. The systematic observations for the study of the
variation of latitude were begun nearly 30 years ago, under the direction
of the International Geodetic Association. Since the world war the
countries in which the several stations are located are financing the
operations involved; but the program followed is laid out by a joint
committee of the International Astronomical Union and the International
Geodetic and Geophysical Union.
The advantage of having, the variation of latitude stations approxi-
mately in the same latitude is that the same stars can be used at the
several observatories and thus the effect of the uncertainties in the star
positions can be eliminated from the results. The probable error of the
determination of the latitude at one of these stations from a single night's
observing is seldom greater than 0".03. The Talcott method is used at
these observatories, with a zenith telescope somewhat larger than the
one generally employed in geodetic astronomy.
Lor~g~tude. Since the diff'eren,ce of longitude between two points can
be expressed in angular measurement and also as the difference in the
local times of the two places, the clet,ermination of the longitude of a
point consists, essentially, in the determination of its local time and
the comparison of a timepiece set to that local time with the timepiece
at a point whose longitude and error of timepiece are already known.
Prior to the invention of the telegraph it was impossible to determine
a difference of longitude with a high degree of accuracy. Observations
of lunar distances and the occultations of stars, were two methods fre-
quently resorted to but both involved laborious calculations and gave no
great accuracy. The met,hocl most commonly employed was to transport
chronometers from a place whose longitude was letdown to the point
whose longitude was to be determinecl. Variations in the rates of the
chronometers, due to motions incidental to transport and to varying
temperature conditions, Tendered this method also subject to large errors.
The more accurate local time observations are now made with the
portable astronomical transit shown in Figure 2, mounted on a low pier
of wood or other material and shielded from the direct rays of the sun
and from the wind by a canvas tent or by a temporary wooden structure.
The instrument is leveled and then placed so that its telescope swings
in the meridian, using, the methods ordinarily employed on such work.
Formerly, a time observation consisted in recording the time of the pas-
sage of a star across each of' ~ number of vertical " wires " in the field of
view of the telescope. The earliest method, letdown as the eve and ear
method, required the observer to leek mental count of' the time by the beat
OCR for page 226
220
FI G URE OF THE EAR TH
of a chronometer and note the time of passage of the star across each wire.
Later a chronograph was used and a telegraph key, connected in all
electric circuit with the chrono~rapl~ pen, was pressed hit the observer
as the star crossecl each wire, thus making a break in the pen trace on
the chronograph sheet. Each of these methods is subject to an uncertainty
owing to the difference in the personal equations of the observers.
During the past few decades what is called the impersonal micrometer,
attached to the eye end of' the telescope, has been used in exact time
observations. i'\ driving mechanism causes a vertical wire to move across
the field of view. As the star comes into the field the movable wire is
placed directly over the image of the star. As the star crosses the field,
the wire is kept or the star either by manual operation of' the driving
mechanism or by an electric drive the speed of which can be regulated
by the observer. As the wire is moved, ~ number of electrical contacts are
made in the apparatus which cause a record to be made on the chrono-
~raph sheet, on which are also recorded the second beats of the local
clocl; or chronometer. The automatic breaks made by the impersonal
micrometer correspond to the breal~s caused by the telegraph key in the
leek method but they are practically free from any effect of the observer's
personal equation. It its readily: seen that the chronometer time of' transit
of the star can be accurately sca.lecl from the chronograph record.
~ determination of' the clocl: correction usually depends on observa-
tions on five or six stars, some to the north and some to the south of' the
zenith of' the point of observation. The methods employed are designed
to enable one to compute corrections for instrumental errors. The striding
level is read during the observations in order to determine and make
corrections for the inclination of' the horizontal axis of' the instrument.
The probable error of the determination of the error of the local clocl: is
usually less than 0.03 second oft' time. During each ~:~ight's work two
determinations of the clock error are made in order to learn the rate of
the clock between the two time sets.
NVith the advent of the telegraph, the difference in longitude between
two places could be obtained with great accuracy, for the known and
the unknown stations could be connected by telegraph and the clocks or
chronometers at the two places could be easily and accurately compared,
thus enabling, the observers to obtain the difference in the local times of'
the two points, and consequently the difference in the longitucles. The
time required for the transmission of' the signals was eliminated by hav-
ing part of the signals sent in one direction and part in the other. There
are many cases in the work of the Coast and Geodetic Survey in which
two observers determined the difference of longitude between each two
of a number of stations form.in~, a circuit.. The accuracy of the longitude
OCR for page 227
DETERMINATION OF GEOGRAPHIC POSITIONS 227
Fig. 3. Theodolite for first-order triangulation. The 9-inch horizontal circle
is read by each of two micrometers to the nearest second of arc.
OCR for page 228
228
FI G USE OF Tl-~E EAR TH
determinations is indicated by the correction per difference to close the
circuit, which was seldom greater than 0.01 second of' time.
Shortly after the world war, the observers of the Coast and Geodetic
Survey began to use the radio time signals sent from the Naval Observa-
tory in TVashin~ton, D. C. An apparatus for receiving the signals was
designed by Drs. E. A. Eckhardt and J. C. Marcher at that time mem-
bers of the Bureau of' Standarcls. NVith this apparatus radio signals were
recorded automatically by the pen of' the chronograph on the same sheet
as was then used in recording the breaks of the local clock. By this
means the direct comparisons could be made of the Naval Observatory
time and the local time of the field station. Experiments were made to
determine the lag of the receiving apparatus, and the Naval Observatory
furnished to the field observers corrections to the time signals sent out
from the observatory. It is not known definitely what is the maximum
error which may enter into a difference of' longitude determined by radio
signals but it is believed that it is little, if' any, greater than the error
involved in the determination by the wire telegraph. The great advantage
of the radio time signals in longitude work consists in the elimination
of' one field observing unit and the flexibility of the method. Previously
the field longitude stations had to be located alongside a telegraph line
or a special wire had to be run to the field observatory. With the radio
method a lon~itucle station can be located at any point to which the
instruments can be transported.
Errors dne -to ast~-o~rwmical de-term~na-t~ons of geog'~aphiopos~t~ons.
NA7hile it is possible, by observations on the stars, to determine the astro-
nomical latitude and longitude of' a point within about 20 or 30 -feet, yet
the position thus determined is of small value by itself in accurate survey-
in~ and mapping operations. The theoretical vertical axis of the astro-
nomical instrument has the direction of' the plumb line at the point of
observation, and the observations on the stars are referred to that line.
Gravity, as is well letdown, acts at right angles to the water or equipotential
surface at the point of observation. This surface is not a mathematical
one, but is quite irregular. It is tilted upwards in the vicinity of high
lards, such as plateaus and mountains, and has the reverse tilt as valleys
and bodies of tidal water are approached. If we use the ellipsoid, or
mathematical surface, which may be considered as closely approximating
the mean water surl'a.ce or Void surface of the earth, we shall find that
the actual direction of' crravit.y deviates in amounts varying up to a
tribute from the normal to the ellipsoid at the same place. It will be
seen front the above that the distance and direction between two points
cannot be determined accurately from their observed astronomical lati
, v v
OCR for page 229
DETERlllINATION OF GEOGRAPHIC POSITIONS 229
tudes and longitudes. The difference between the direction of gravity
and the normal to the ellipsoid at a point is called the deflection of the
vertical.
There are, on record, many cases of decided deflections of the vertical
which would be.very troublesome to the surveyor and map maker should
he be unaware of their existence. On the island of Porto Rico there are
astronomical latitude stations at Ponce, on the south coast, and at San
Juan on the north coast. The distance between those two observatories
was determined by triangulation and found to be about one mile shorter
than the distance computed from the astronomical latitudes. Several very
large deflections of the vertical have been found near the great mountain
masses in the western part of the United States, and in Asia also.
It was realized at a comparatively early date that accurate surveys
of large areas could not depend upon astronomical latitudes and longi-
tudes alone; but in many cases no other method was available and much
confusion has resulted. Notable examples are found in the location of
many of our state boundaries., which are frequently defined by law as being
located on a certain parallel or meridian. The astronomical location of
that parallel or meridian might, because of deflections of the vertical, be
as much as one-half mile from the designated latitude or longitude as
referred to the ellipsoid. Moreover, since the dedections would vary in
amount from one astronomical station to the next, the boundary would
follow a zig-zag course. The eastern boundary of Montana is at one
place more than one-half mile to the eastward of the position it would
have if referred to the ellipsoid, and the State of Kansas alone, the 98th
meridian is one-fourth mile wider than the north and south distance
between its designated boundary parallels on the mathematical surface.
As a matter of fact, no legal doubt is thrown on a boundary located
astronomically, for a boundary once accepted by authorized methods
remains fixed as located, but it is confusing for a boundary line to pursue
a rambling course across a map at a considerable distance from its nomi-
nal parallel or meridian.
GEOGRAPHIC POSITIONS DETERMINED BY GEODETIC METHODS
The errors inherent in Fair positions by astronomical methods are ob-
viated by triangulation. A series of points several miles apart are selected,
so located that the lines joining intervisible pairs of points will form a
series of connected triangles. A side of one of the triangles is measured
accurately, and the angles at the vertices of the triangles are also measured.
The measured length and observed angles furnish sufficient data for
computing the lengths of the other triangle sides; and thus the distances
between points become known.
OCR for page 230
230
FI CURE OF THE EAR TH
In any triangulat,io~1 system the latitu`:le and longitude of some initial
point must be determined by astronomical methods, and also the direction,
or azimuth, of' some tria.:r~gle side radiating from the initial point to
an adjacent station must be determined astronomically. If then the shape
and size of' the ellipsoid surface most closely coinciding with the mean
geoid surface is known, the latitude and longitude of each point in the
triangulation scheme can be computed.
In beginning the Napping of any large area, such as that of the
United States it is inevitable that there must be for a time many detached
triangulation systems, each based on one or more stations at which the
astronomical latitudes and longitudes have been determined. Later the
several systems of triangulation will join and then it, will be necessary
to eliminate the gaps, overlaps or offsets which exist between each. two
separate systems. All discrepancies are eventually eliminated by having
a single point in the country as the initial or datum to which all other
stations are referred. The datum for the United States is based on
station Meades Ranch in central Kansas. This point was selected be-
cause it was near the center of area. of the United States and because
it was common to two great arcs of triangulation extending across the
country, one along the 39th parallel of latitude and the other along the
98th meridian. A latitude and longitude were computed for the station
Meades Ranch which made the sum of the squares of the differences
between the astronomical latitudes and longitudes and the triangula-
tion latitudes and longitudes a minimum. This computed value was so
near the value obtained by basing the position on the datum previously
ire use in New England that the latter value was adopted to save the
labor of re-computing all the triangulation of the northeastern part of
the country.
After the adoption of the standard datum for the United States, con-
nections were made between this system of triangulation and the systems
of Canada and Mexico. When this had been done those countries decided
to adopt the same datum. based on Meades Uncle as used in the United
States, and hence the geographic positions for the three countries are
all said to be based on the North American datum. This is the only
continent having all of its triangulation referred to a single ellipsoid
and a single initial point.
To determine geographic positions by geodetic methods or t,riangula-
tion three classes of operations are necessary, as has already been indi
cated; viz., base measurement., angle measures, and observations for
azimuth. Each of' these operations will now be described.
Base measuremerl,t. A base line may now be measured with almost
any desired degree of accuracy, but this condition hats existed only during
OCR for page 231
DETERMINATION OF GEOGRAPHIC POSITIONS 231
the past four decades. Previously there was great difficulty in measuring
a base line and the best thoughts of geodesists and physicists were given
to devising apparatus and methods for that class of work.
From the early years of triangulation until the latter part of the
nineteenth century, bases were measured with bars of: various metals
and also of wood and glass. No one of these early bar apparatuses, which
are called " single bars," was entirely satisfactory because of the great
difficulty in determining the temperature of the bar during field measures.
This difficulty, however, was largely overcome by the use of what are
called duplex bars in which two rods of metal having different coefficients
of expansion were employed. Noteworthy among these was the Eimbecl:
base bars first used in the measurement of the Great Salt Lake base.
This Eimbeck bar had one rod of steel and another of brass which were
rigidly joined together at one end of the bar. A scale at. the other end
of the rods made it possible to determine the relative changes in the
lengths of the rods for different temperature conditions and from these
v
· , ~ ~ ~ P ~ 1 1 ~ _ __ 1
scale readings the average temperature ot the two rods COU1Ct oe ~erlvea.
In addition there were three thermometers placed in the covering tube
in such a way that the mercury bulbs would be close to the two metal
rods and the corrections to the bar lengths derived from the thermometer
readings afforded a check on the scale readings. The probable error of
the length of the Salt Lake base, as measured by the Eimbeck bars,
was one part in five million. The various bars used in base measurements
had different lengths but the usual one was :hve meters. During the
measurements the bars were successively placed in contact end to encl,
in careful alignment, and corrections were made for the inclination of
each bar from the horizontal.
The iced-bar apparatus was first used by the Coast and Geodetic Survey
in 1891. It is probably the most accurate base measuring apparatus ever
invented. It is still used for laboratory standardizations but is so slow and
so expensive to use that it is no longer employed for field measurements.
In principle it consists of a standardized bar packed in melting ice which
keeps the bar at constant temperature. Micrometer microscopes mounted
on posts serve to :~x the positions o-t the terminal marks and to howl
them while the bar is moved forward to its successive positions along
the base.
About 1887 Jaderin of Denmark made experiments in the use of long
steel wires under a constant tension on the measurement of base lines. He
found that, with proper procedure and under favorable temperature con-
ditions, such as prevail as .~.v ~^ ~^ of ,, ~
accuracy equal to that secured by the best bars. The engineers of the
Coast and Geodetic Survey followed with interest Jaderin~s work and
at night. fir on clo~r Anv~ to nn331d nht.nin n.n
OCR for page 232
232
FI GTJPE OF THE EAR TH
tested his method in 1891 on the Holton base in Indiana, using steel
tapes instead of wires. ~ number of precise base lines were measured in
this country with satisfactory results, by using, 50- or lOO-meter steel
tapes.
In the early part of the present century the steel tapes and wires used
on base measurement were superseded by those made of invar, which
is an alloy of nickel and steel first made by Prof. Guillaume. Invar, as
is well letdown, has a very low coefficient of expansion, the value varying
with the relative percentages of nickel and steel and with the metallurgical
treatment ¢,iven. The coefficient of expansion is usually less than one-
tenth that of steel and sometimes is practically zero or even negative.
Experiments made in 190~; by the Coast and Geodetic Survey in measure-
ments of bases with invar tapes proved that measurements could be made
satisfactorily even in sunshine.
There is only one difficulty in using, invar tapes or wires in precise
base measurements and that is due to the instability in length of the
invar alloy. The tapes become more stable, however, when they are a
few years old, especially if they are handled carefully, and the effect
of changes is largely eliminated by standardizing, the tapes immediately
before and after measurements in the field. These standardizations are
made at the Bureau of Standards where a 5-meter iced-bar is used as
an intermediate standard between the prototype meter and the tape to
be standardized.
Field measurements. After a site has been selected and cleared of
brush and other obstructions, stakes are driven at bO-meter intervals
along. the line of the base, and strips of copper or other soft metal are
fastened to the tops of the stakes and aligned between the base ends.
At the beginning of the measurement, one end of the tape is held directly
over the metal tablet marling the end of the base and when the proper
tension has been applied the mark on the forward end of the tape is
transferred to the metal strip fastened to the top of the stake at the
forward end of the tape. Then the tape is moved forward, the graduation
mark at the rear end of the tape is brought into contact with the mark
previously macle on the metal strip and the forward mark on the tape
is again transferred to the adjacent metal strip, the process continuing
along the base.
~ base is divided into kilometer sections, each section being measured
in both the forward and backward directions, with different tapes. At
least three tapes are used on each base, the measurements being so ar-
ranged that each tape is used with each of the other two tapes over
approximately the same distance. An inter-comparison of the three tapes
is thus obtained and if any tape has changed its length appreciably from
OCR for page 233
DETERMINATION OF GEOGRAPHIC POSITIONS 233
that given by its last standardization that fact will be disclosed by com-
parison of the section lengths as derived from measurements with the
different tapes.
So far as possible the held measurements are made with the same
tension and method of support as obtained during standardization and
where this is impossible proper corrections are made for the different
conditions. Two' mercurial thermometers attached to the tape are read
for each tape length and the readings are used to derive the temperature
correction. 'lihe elevations oi the tape supports are obtained by leveling,,
in order that any inclined tape lengths may be reduced to the horizontal.
The measured base Is also reduced to its sea-level length, for the tria.ngu-
lation must be computed as if the base lines and the triangles were
measured on the sea-level surface.
The probable error of the measured length of a base line when invar
tapes are used is generally about one part in two million. The actual error
is believed always to be smaller than one part in 300,000. That accuracy
is 'all- that is needed in the general triangulation net of a. large area.
That a far greater accuracy than is needed in ordinary triangulation
can be obtained in base measurements with invar tapes is shown by the
results obtained in measuring a base line near Pasadena, Calif., in 1924,
as a part of the determination of the distance between Mt. Wilson and
San Antonio Peal:, the line that was used by Prof. A. A. Michelson, in
the determination of the velocity of light. Extreme care was used in the
measurements and eight tapes were employed. Each kilometer section
of the base was measured at least Tour times, each measurement being,
made with a different tape. Extreme care was taken at the Bureau of
Standards in determining the length of the eight tapes used in the fielcl,
both before and after the field measurements, the Bureau certifying that
the probable error of standardization of none of the tapes exceeded 1 part
in 2,000,000. The probable error of the length of the base from, field
measurements and observations was, + 3.40 mm
Since the total measured
length of the base was about 40,048 meters (the projected length being,
~ ~ v ~ _
33,638 meters), the probable error of its measurement was one part in
11,600,000.
The triangulation used in connecting the measured base with the line
between hit. Wilson and San Antonio Peak was executed with unusual
accuracy. Even the effects of the deflections of the vertical on the values
of the measured horizontal angles at the several triangulation stations
were determined and corrections therefor applied. The air-line distance
between the two mountain stations is 3d,385.53 meters and the probable
error of this value is ~ part in 6,80~0,000. It is believed that the actual
error in this distance is not greater than one part in 1,000,000. Brief
OCR for page 234
234
FIGURE OF THE EARTH
accounts of the measurement of the distance between Mt. Wilson and Sail
Antonio Peak and the measurement of' the velocity of' lights are contained
in the Astrophysical Journal, vol. Go, pi. 1-22, 1927.
Argyle release Rents. Theodolites of various sizes and degrees of
excellence are used to measure the angles of' triangulation, though on
hrst-order triangulation only the very best instruments are employed.
(See Figure 3.) A theodolite, in its essential principles, is similar to
the surveyor's transit with which ever,: one is I'amilia.r. The main di~er-
ences are in the refinement of workmanship in fitting, the various parts
together, especially the vertical axis, in the ,~,ra,duation of the circle, in
the high quality of the optical parts, and in the use of micrometer micro-
scopes by which readings of' the circle can be made to single seconds
of arc.
Until recent years it was believed that large hori%ont,al circles were
necessary in precision theodolites, but with the increased perfection of
" graduating engines " smaller circles have come into general use. The
modern theodolite usually has a circle ranging between 8 and 9 inches
in diameter and there is one first-order t.heodolite of' a special design
which has a circle less than 6 inches in diameter. The theodolites generally
used have two micrometer microscopes placed 180 degrees apart, the
mean of the two readings being free from the effect of eccentricity of
centers. One measure of' a direction to an adjacent triangulation station
consists of two pointings ol' the instrument, one with the telescope direct
and one with it reversed in tl~e woes, and of rea.clin:,s of' the horizontal
circle for both pointin~,s. In observing,, the telescope is pointed and the
circle read on each station in turn. around the horizon in a clockwise
direction beginning with the initial station, and then the telescope. is
reversed in the wyes and the paintings and reaclin~,s are made in the
reverse order back to the initial station. On first-order triangulation
16 of' these measures are usually made and the mean is taken as the
values for the several directions to the adjoining stations. After each
measure the circle is shifted to a new position, thus ma,ki~¢, the mi-
cromet.er readings I'a,ll on di-ff'erent parts of' the circle. By doing this the
effect of the systematic errors in the graduation of' the circle on the
mean value of' an angle is largely eliminated. The probable error of a
direction resulting front the adjustment of a triangulation net is usually
about +0"~.
The theodolite is mounted on a wooden stand where the stations are
located on mountain peaks or sharp hills or ridges; where the stations
are in level country the instrument is mounted on a tower which is used
to overcome the effect of the curvature of the earth and to lift, the lines
above obstructions such as trees or buildings. ~ See Figure 4. ~ The
OCR for page 235
~70\ OF 6~0~76 F037~3 233
FIG. 4.~ri~nguJ~i~ to~er~his steel tower can
be quickly tacked down, moved by ~ Eagle tack to
^ 1~, ~ ~ ~ ~ ~ Ha.
Tbe inner tower support the tbeodobte and the
outer one the obs~viD~ platform and the ~1 light
~bicb is Baboon to other stations. Tbe two Myers are
entirely separated from each other.
OCR for page 236
236
FIGURE OF TI-IE EARTH
objects s;~,htecl on arc usually lamps or heliotropes, the former being, used
for n.;~,ht observations and the latter cluring days when the sun is
, · .
snlnln~
Azimuth o19servatir)ns. There is a tendency for an arc of' triangulation
to swerve from its true position to the right or left of the general direction
of progress. This is due to an accumulation of observational errors a,ncl
also probably to meteorological influences such as the direction of pre-
vailin`~ winds. This swerving is controlled by the use of what are called
Laplace azimuths, located at intervals of six to eight triangles along the
scheme of t,rian¢,ulation. A Laplace azimuth is the astronomic azimuth
of' a line between two triangulation stations corrected for the effect of
the deflection of the vertical at the point of observation. The astronomic
azimuth is obtained by measurin,, the horizontal angle between Polaris
an:1 a triangulation station and correcting the measured angle by the
angular distance of Polaris from true north at the time of' observation.
Cla.ss~hTcat~on of tri.angq~lc`.tion,.- Triangulation is classed as first, second
or third order to correspond with the varying degrees of accuracy with
which the lengths and directions of the lines between the stations are
cleter~ninecl. On first-order triangulation the adjusted distances should
be correct within 1 part in 25,000, on seconcl-order to within ~ part in
10 000 awl on third-order to within ~ part in .5,000. The general plan
being, following, in the triangulation cont,ro1 of the surveys and maps
of the United States is to provide that no point in the country will be
further than about 25 miles from a triangulation station located with
-first- or second-order accuracy-. Intermediate areas will be covered by tri
,, ~
an`~ula,tion of thircl-orcler accuracy for the immediate control of detailed
surveys.
Since the object of triangulation is to determine the geographic po-
sitions of points on the earth's surface, and the distances and directions
between them, for the use of engineers, map makers and others, it is
essential that the determined points be perpetuated. This is done by
placing, inscribed metal tablets in blocks of concrete or in outcropping,
lock. (See Figure 6.) Accurate descriptions of the various stations are
written and published for the use of those who may need the data.
The first precise triangulation clone in the IJnited States was in 1816
in connection with charting the coast in the vicinity of Long Island
Sound.. This worl: has been cont,inuecl since then and now there are
2G,OOO miles of triangulation arcs in this country. (See Figure 6.) The
IJ. S. Coast and Geodetic Survey, in cooperation with the Geodetic Survev
of Canada, has also extenclecl an are of triangulation, from the north-
western part of our country to Skyway, at the head of Lynn Canal, in
Southeast Alaska. Eventually that are will be extended into western
OCR for page 237
~70\ OF ~0~C [03~70~S 237
Alaska RD] to Bering Sea. A sechon of an ~C of triRD#l~ion DO Ibid
is also ~ base bled indicated in heavy lines; is sin in Figure ?.
~~a~6 ~/ r86~z76. II1 older abut the results of tri~Dgul~tion may
be Bade isle as the Bock progresses; bed arcs must Be ages into
the old ones; vita the older geographic Horizons bel] axed. Ibis process
usually continues until such time as ~ large area bus been r~tber co~-
pletel~ covered Aim ~ Ir~e-wolk of trisn~l~tion arcs. Wbe~ this has
been ~ccomplisbeJ it is desirable that ~ readjustment of ~ triangulation
net he Dudes in order that the tbeoretic~lly Best geographic positions
may he Stained IOT the sf~tions. ~ few peats ago the system of arcs Tar]
~:~
FIG. 3~DGu19tiOO static tuLIet.-A bronze twist With BD ~pproph~te
inscription is sot in rock or ~ concrete monument to mark prominently the
exact site of ~ ~i~ul~tion Station.
ace extended over Me Catch belt of the Waited States to such an
extent that ~ hIl~l ~4jus~ent could he ~de. Ibis Ads done ty ~ method
devise] By Facials of the Coast ~] geodetic Survey ~bicb prove] to De
entire s~tisI~o~ for the purpose. Ibe Hotbed c~pIoyc] ~] the
results icri~cd R1C ~cscritc] in Specie Publication No. 139 of the Coast
] Ocodetic Survey.
Ibc accq~cy Aim which f~st-o~]er t~i~l~tio~ is gone is cleric
indict fly the closing errors of the loops ~ t~i~ngul~ou in We Weston
part of We counting i~Ql~c] in the [C]]jUSt~C~t. (See BTgur~ S.) Ibe
closing errors of We 16 loops of the get area in ~11 c~scs, less task one
part in 130~000 ~ the distance RI-D] We loop tbrou~ We ages of the
OCR for page 238
238 FIGURE OF Y7HE EARY7H
.
~!
''I . ~
. ~
\ (1
~ \
~,,'
-''i ~
it_
c
C ~)
CN'
Be, _
~ ~7
Q}
.
o
o
.~
:t
_
1 .~
so
a:
o
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CO
~3
OCR for page 239
DETERMINATION OF GEOGRAPHIC POSITIONS 239
triangulation arcs. It will be noticed on Figure 8 that there are only
two circuits which have closing, errors greater than one part in 200,000
and the average correction for the 16 loops is one part in 460,000. It
is rather remarkable that the closing error for the perimetric circuit of
6,300 miles is only 33 feet or one part in 84S,000.
It is expected that, by the end of 1930, the triangulation net coverin,
the eastern half of the United States will be ready for a: readjustment
similar to that which was made for the western half of the country.
l
6rrrecn<~ tow
Ail / ~
\~ I H - a Bate
5'.~\;~:
.~
,,/
1 sac_ ~i=
1 _
/
/
Sate b110-,~
7:orrect
how
cue of s Jr
AL 10
}~etrc6
! ~1~1 1 1 it__
0 5 10 ~20 2b 30 3
Fit. 7.-Triangulation scheme connecting with a base line.
Sc~er~t~;fic uses of triangulattorv. Although the main purpose of tri-
angulation is utilitarian, largely for survevin~ and mapping, yet there
are important scientific uses made of it, especially of that observed with
fir.st-order accuracy. The first of these uses is to determine the shape
and size of the earth. By comparison of astronomical positions with the
positions of the same points determined by triangulation, data are se-
cured for use in obtaining a closer approximation to the true dimensions
and shape of the ellipsoid surface which most nearly coincides with the
geoid or sea-level surface.
16
OCR for page 240
240
FIGURE OF 'THE EARTH
Secol~dl,v, trian~,ula.tion clata, when compared with astronomical data,
make it possible to determine the deflection of the vertical. This deflec-
tion at any point is due to the influence of the irregular surface of the
earth and to the variable densities of' the earth's crust. There are areas
above the sea-level surface and other areas which are depressed below it.
These areas have excesses and deficiencies of' mass, respectively, which
deflect the plumb line, or direction of gravity, from the line that is
normal to the ellipsoid surface. It has been found that the cleflcctio~s
.. ,,,~ . ... ~ ~ .0
''~1
At/ TO-/ ~ 29-7700 1 ~ /.-848,000:
~r
egend_ Feet_ les ,
Proportion ~ ~1
--~~'~'': l
1
j)-~ /-- 848, °°°~
i- -) / 3s4,000t . ..
He'' 1 ~ ~ ;
r ~,,:
~ _ '1
.P
/ 1~= ~/! 0560 1 Hi- /3-001 i, 5 ,350 ~
L:\f~=
id, :35 - /330
I.. . / .200,~
Hi/ ---I...!
~ ~ / '-'5~1,~1 ~29 - /700
Legend-
1= ~ :'~
Feet - Miles 4~~
Proportion '2 - to
i\
FIG. 8. Loop closures resulting from read ustment.-The
first number above the line is the total closure in feet and the
second number the approximate length of the loop in miles.
Below the line is the approximate proportional part of the
whole circuit represented by the closure.
are not as great as would be caused by the excesses or deficiencies of mass
referred to. An outcome of the researches involving the deflection of the
vertical and values of gravity has been the proof of the theory advanced
three-quarters of a century ago to the e-elect that continental and island
areas are underlaid by crust,al material that is lighter than the average
and that tidal water areas are underlaid by material denser than normal.
This balancing of the irregular distribution of' material at the earth's
surface has been called isostasy.*
* See References.
OCR for page 241
\ 0~ 0~0~O F0~70~S 241
~ Bind USC of tri~l~tion is to Je~r~iD~ talc extcDt of bo~izo~t~1
couth mo~cmcnts in regions of scTsmic ~rtivit~ Curing an keg or
during the intcr~1 Lctv~cn e~rthqU~Lcs ~bcn go crust~1 m~teTi~1 is
under stress. In talc Laity States thcrc bus tack ~n c~tcnsi~c in~c~ti
~tion of this kin] in California; with ~ airy to disclosing Chat mQ~c-
mcnts bloc taken ~l~cc Curing Me past qu~rtcr CCD61~. The results of
,
this in~cstig~tion are given in ~ recant report of the Coast and Gcoictic
Survey.
For ~ cc~tury or more steady progress bus been Bare in the processes
of Jetc~mining geographic positions hy ~st~oro~ic~1 and gooJetic ~eth-
offs. Abe instruments for Said Fork hays aced g~e~tIy improved; and ~lerc
bare also Been rubric] JeveIop~ents in geld and ounce methods. ~ large
part of the rapid Jevelopmcnt of geodetic ~strQ~omy; triangulation; and
base measurement is due to the conferences and publications of the Inter-
n~Iion~1 OeoJetic Association and its successor, after the ~oTl] cart the
Action of Geodesy of the Intern tin ~codetic and Geo~bysic~1 Anion.
At the trienni~1 congresses; reports of member countries are p[csente]
and Jiscussed; and the delegator are gale to mono ~erson~1 contacts Rich
lead to ~ uniorst~4i~g By the geoJesists of ebb country of the Fork
~ ID ~ the opera
Bowing WiDium. Determination of fume, longitude, latitude and azimuth (5~
Edj. lo. S. Coast Ceod. anon Spec. fitly. ~0. 14:1-177 (1913). DescHhes
methods and instruments used on geodetic ustroDonly in the Coast and
Ocodetic Survey.
- . Isostusy. lo. F. Button Aid Co., \~ York, p. i-xiv and 1-273 (1927).
Contain ~ene~1 e~pl~hoD of the prindplo of imposts.
- . Andy of time errors in precut longitude determinations by the C. S.
Coast and Ceodetic Survey. C. S. Coast Mood. Surv. Spec. Pub. Do. 92: 1-9
(1923). Discusses the errors in determining time with held types of ustro
Domic~I transit in Coast and Ccodetic Survcy and the closures obtuincd in
longitude loops aside metric Dire circuits.
Clarke, Agnes hi. A popuI~r bights of astronomy during Me 19tb century.
(4tb Ed.) A. and C. Black. London, p. i-xv and 1~9 (19~). Includes also
~ r~tber comprebenEive account of the bcgiDnings of ~tronomic~1 science.
Comic Cede D., ~ E~b~rd~ F. A. Wird~ longitude. U. S. Coast Good.
Surv. Spec. Pub. Do. 109: I-lY, 1-S2 (1924). DrscHb~s the apparatus and
methods used iD the radio dct~minutio) of ~ precise diJercnce of longitude.
Encyolop~di~ B~t~nnicu. Articles on astronomy and geodesy.
Core, J. Bollard. Ceodesy. Uou~bton, Origin and Co. Boston and Bed Yolk,
a. Phi and 1-218 (1891). A brief accost of the various dcterminutions of the
~z~ add sb~pe of the Crib Huh descriptions of methods used in dctcrminu
. . . .
boa of gcogruphlc pos~lons.
* Special Publication Do. 151.
OCR for page 242
242
FIGURE OF THE EARTH
Hodgson, C. V. Manual of first-order triangulation. U. S. Coast Geod. Surv. Spec.
Pub. No. 120: I-VI, 1-185 (1926). Describes in detail the operations in field
and office for determining and computing first-order geographic positions
geodetically by triangulation method.
--- . Manual of second and third order triangulation and traverse. U. S. C.oast
Geod. Surv. Spec. Pub. No. 145: I-V, 1-226 (1929) . (covers the field for second-
and third-order geographic positions that Special Publication No. 120 covers
for first-order work.
Hosmer, George L. Geodesy. (2nd ea.) John Wiley and Sons, Inc. New York,
p. i-xiv and 1-461 ( 1930) . A statement and discussion of the principles of
geodesy and of geodetic surveying.
. Textbook of practical astronomy. (3rd ea.) John Wiley and Sons, Inc.,
New York, p. i-xi and 1-270 (1925). Presents theory and practice of field
astronomy from the standpoint of the engineer.
Royal Geographical Society. Hints to travelers, scientific and general. Vol. II.
(lOth ea.) The Royal Geographical Society, London, p. i-ix and 1-318 (1921).
Discusses methods and instruments used to determine latitude, longitude 'and
azimuth on exploratory surveys with specimen observations and computations.
Representative terms from entire chapter:
probable error
