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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

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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

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DETERMINATION OF GEOGRAPHIC POSITIONS 223 F'IG. 1.-Zenith telescope. An instrument designed especially for determining astronomical latitudes by the Talcott method. 15

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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

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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

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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

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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.

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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

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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.

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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

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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

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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

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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

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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

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~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.

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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

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~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

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238 FIGURE OF Y7HE EARY7H . ~! ''I . ~ . ~ \ (1 ~ \ ~,,' -''i ~ it_ c C ~) CN' Be, _ ~ ~7 Q} . o o .~ :t _ 1 .~ so a: o 1. CO ~3

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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

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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.

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\ 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.

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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.