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Geodesy is one of the oldest of the geophysical sciences. Originally
it embraced only a limited field, consisting of the determination of the
shape of the earth and its size and the means by which this was done.
But now the subject of geodesy is very closely allied with other matters
and there are few geophysical questions that do not, either directly or
indirectly, concern the geodesist in the solution of his problems.

* Published in the Proceedings of the National Academy of Sciences, 6, 1920 (545-601).

Almost every country of the world that pretends to any standing has organized a service for the purpose of carrying on what are called geodetic surveys. As a matter of fact, any survey of the land in which the shape and size of the earth are taken into consideration can be called geodetic; thus, the hydrographic surveys along the coast made for a sailing chart are really geodetic surveys and, similarly, a topographic survey of a large area may be considered to be a geodetic survey; but what I have in mind in speaking of geodetic surveys, are triangulation, base measurements, precise leveling, etc.

By means of base measurements and triangulation, the geographic positions of places are determined which are of immense value in the permanent establishment of boundary lines between nations and political sub-divisions of a nation, for the control of various classes of surveys and maps, and for other engineering purposes.

Practically the whole of Europe has been covered by a system of points whose latitudes and longitudes have been determined by triangulation. This is also the case in Japan and in India. Australia, portions of Africa, and some of the countries of Central and South America have made good starts towards extending triangulation over their areas. In the United States excellent progress has been made in this work. There is triangulation around the entire border and there are many arcs in the interior of the country from which triangulation of precise or lower grades may be extended into every area in which detailed surveying and mapping operations are to be carried on.

It will be of interest to outline briefly the methods for carrying on triangulation or trigonometric surveys.

In the first place a spheroid of reference must be adopted which will approximate the actual shape and size of the earth. Then there must be an initial point whose astronomic latitude and longitude have been observed. Next is needed the azimuth or true bearing of a line of which the station occupied forms one end. After these data are available it is necessary to measure with extreme accuracy the distance between two intervisible points on the earth's surface to serve as a base. After the length of the base is known, additional stations are selected with a view to forming triangles by lines which are clear of obstructions and thus intervisible between their ends. It is, of course, a well known mathematical principle that when a side and the angles of a triangle are known, the other sides can be computed. When these sides have been computed from the base, either one of them can be used as the base for a new triangle. In a similar manner we may extend the computations through a long chain of triangles extending in some cases hundreds of miles across a country. When the base has been measured and the angles observed and the computations of the triangles have given the lengths of the sides of the triangles, then computations can be made which will give the latitude and

longitude of each point and the bearing of each line based on the data for the initial point, at which the astronomic latitude and longitude were observed and the true bearing determined of one line of the triangulation. This is the simplest case of a geodetic survey where the whole net work depends on some initial astronomic station.

In practice, we have a much more complicated situation. Geodetic surveys have been started from many widely separated astronomic stations and the triangulation based on each has been finally extended until it meets the triangulation nets based on other astronomic stations. When this has been done, it is always found that the geographic positions computed from one astronomic station will differ from the geographic positions of the same points computed from other astronomic stations. We have a very striking example of this on the island of Porto Rico. The astronomic latitude was determined at Ponce, on the southern coast, and at San Juan on the northern coast of the island. The distance across the island, in a north and south direction, between these stations, is approximately thirty miles. The distance between the two stations, as computed from their observed astronomic positions, differs by about one mile from the distance between the two as determined by triangulation. The triangulation distance is correct within ten feet. The cause of this difference in distance as determined by the two methods is the attraction of the mountain mass, forming the Island of Porto Rico, on the plumb line to which the astronomic observations are referred. To the northward of Porto Rico are the deepest parts of the Atlantic Ocean and just to the southward of the island is a vast depth in the Caribbean sea. It can be readily seen that the effect of this distribution of mass in the oceans and in the island and its base is to draw the plumb line towards the centre of the island. This would throw the zenith on the south shore too far to the south and the zenith at the north shore station too far north. The plumb lines projected against the celestial sphere would therefore include a greater arc than they would under normal conditions. This angle is actually about 56 seconds of arc greater than normal.

In carrying on a geodetic survey, as described above, it is necessary to know accurately the elevations of the base lines and triangulation stations above the so-called plane of reference which is mean sea level. Each line that is determined by the triangulation is referred to sea level, that is, the computed distance between any two stations is the distance, along the geoid or sea level surface, between the points where the plumb lines at the two stations pierce this surface. Accurate elevations are also needed in many phases of surveying, in mapping, and in other engineering operations. They are also essential in many branches of science, notably in meteorology, geology, etc.

In order that elevations may have the maximum value to engineers it is necessary that they all be referred to the same, so-called, plane of

reference. The one which has been almost universally adopted is mean sea level. It may be defined as the surface which would coincide with the surface of the oceans and their tide water branches if the tide producing forces should cease to act and there were no movements of the air and the barometric pressure were uniform. This surface can be determined at any point on the exposed coast by continuous tidal observations for approximately three years. The mean of the hourly readings of the tide record for such a period is practically free from error. If observations were made for a long period of years, it would be found that the average position of the surface of the ocean, for one three-year period, would agree very closely with the position of the surface for any other three-year period. The difference, if any, would be very minute.

It has been assumed by geodesists that, insofar as engineering work is concerned, mean sea level at each place along an open coast is in the same equipotential surface. This, of course, cannot be absolutely true because of different densities in the waters of the oceans, different barometric pressures, trade winds, and possibly other causes, but the deviation of mean sea level at one station from the equipotential surface containing mean sea level as determined at another station is so small as to be negligible in surveying, mapping, and other engineering work, and for practically all scientific purposes.

After the mean sea level surface has been determined, lines of very accurate spirit leveling are extended inland from the fundamental tidal stations. Each country of the world has done more or less of this work and in the United States we have tens of thousands of miles of this grade of leveling.

When a network of triangulation has been extended over a large area and astronomical latitudes, longitudes and azimuths have been observed at many of the stations, differences between the astronomic and geodetic positions of the stations are found. These differences are due to the deflection of the vertical, as was explained in the case of Porto Rico, and to the fact that the spheroid of reference used in computing the geographic positions of the triangulation stations differs from the actual figure of the earth. The former differences, namely, those due to deflections of the vertical, tend to be more or less accidental in character, while those due to an erroneous spheroid of reference are systematic.

After the triangulation and astronomic work have been done over an extensive area, the shape and size of the mean figure of the earth may be computed with a higher degree of accuracy, possibly, than the spheroid of reference on which the triangulation was originally based. For one hundred years or more we have made closer and closer approximations to the true figure of the earth, as the triangulations of the various nations have been extended. In the earlier of these determinations there was considerable uncertainty, owing to the deflection of the vertical caused by

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