EDGE OF THE EARTH'S SHAPE AND SIZE, T By C. A. SCHOTT HE Survey has just published a quarto volume containing an account of the transcontinental triangulations and measurements of an arc of the parallel in latitude 39°. It also has ready for publication the manuscript giving the result of an oblique arc in the eastern part of the United States. Both are contributions of great length and among the first of their kind in America. At first sight it might appear rather late in the history of the Survey to bring out results of the earth's figure. But it But it should be remembered that such measures were not the prime object of the Survey in its early stages of activity, but came about in the natural course of continuous development during nearly two-thirds of a century. What was required was to secure a series of geodetic measures consistent within themselves and serving as a bond binding together the separate detail surveys so as to form ultimately a systematic whole. This requirement demanded the establishment of extended primary triangulations not only along our coasts, but also as a connecting link across the country from ocean to ocean, to secure uniformity of results. The growth of these operations depended of course upon the immediate requirements of the Survey for the production of harbor and coast charts and was subject to the means available from year to year. Thus when I state that the first was made between the years 1844 and 1898, and the second between the years 1833 and 1898, I do not mean that it took fiftyfour and sixty-five years respectively to complete the task. Indeed, in either case there were many years of interruptions. What these long intervals signify and emphasize, is, that the various operations of the Survey were more urgently required in the production of practical results for immediate use and in a great measure as aid to navigation. For the more technical part of the work the available knowledge of the earth's magnitude was sufficient for the early needs. At the same time it was recognized that the measurement of the earth required the same means and methods as that of an extended country, viz., a net-work of primary triangulations and a number of astronomical determinations for latitude, longitude, and azimuth of its points. In time, therefore, sufficient material would accumulate to direct special attention to this, the highest feature of geodesy. After the triangulations had reached hundreds of miles in extent, and the geographical positions had been determined by their development upon the surface of a spheroid representing the shape and size of the earth, it became a matter of importance to see that the direct astronomical measures for latitude and longitude kept in close accord with the corresponding geodetic measures; thus it came about that in February, 1880, the Survey changed its first reference spheroid, that of Bessel, for a more suitable one, that of Clarke of 1866. When in 1889 the United States, by resolution of Congress, consented to become a member of the International Geodetic Association for the measurement of the earth, the subject of the measures of arcs came into greater prominence, and thus the field work of the two arcs, then fairly under way, was accelerated and brought to a close late in the year 1898. Before entering upon the detail of the two arcs it may not be out of place to state that in order to obtain a measure of the dimensions of the earth, as represented by a spheroid, that is, by a surface generated by the rotation of an ellipse about its minor axis, it is essential that we should be in possession of at least two arcs or of an equivalent thereof. For combinations of two arcs of the meridian, their mean latitudes should differ widely; the same is true for the combination of two arcs of the parallel. We may also obtain an arc of the meridian with one of the parallel, but in every case the measures should be of considerable extent. Arcs of less than 5° (about 556 km., or 345 st. miles) would now be regarded as short ones. It has been stated that one of our arcs is an oblique arc, and as it possesses a great range of latitude and also of longitude and is supplied with a large number of astronomic measures, it is of itself sufficient for the deduction of values for the dimensions of the earth. Furthermore, it may be remarked that for any relatively small part of the earth's surface an osculating spheroid may be determined, as, for instance, was done for our oblique arc. Such a spheroid has the property that its surface is in best accord, as regards curvature, with the actual or physical one, the latter considered as a mathematical surface of equilibrium and generally known as geoid. The definition of an osculating spheroid thus implies that the sum of the squares of the difference between the various astronomic and geodetic measures be a minimum. The mathematical treatment of the combination of the arc measures differs according to their nature, whether they are extended in a certain direction or whether large areas are covered, but in its generality it is necessarily laborious. The salient points of the two arcs under characteristic of the triangulation is its rigidity imparted to it by quadrilaterals and other polygons. In crossing the Rocky Mountains many of its sides exceed one hundred miles in length, and there is one side reaching to a length of 294 km., or 183 st. miles; the altitude. of many of the stations is also considerable, reaching to 4,300 metres, or 14,108 feet, in the case of Pike's Peak, and to 14,421 feet at Mount Elbert. All geometrical conditions subsisting in the triangulation are satisfied by adjustment, inclusive of the required accord of the base lines, so that the same length for any given line is found no matter from what line one may start. This involved much heavy work; for instance, the triangulation adjustment between the Salina and the El Paso base demanded the simultaneous solution of ninety-nine normal equations (with as many unknowns). In addition the figures required the evolution of a correction to each of the two hundred and twenty-five observed directions. Coming to the astronomical measures, we have distributed over or near the arc one hundred and nine latitude stations, occupied almost exclusively with zenith telescopes; there are, also, seventy-three azimuth stations, various methods having been used, and lastly we have twenty-nine telegraphically determined longitudes. These, of course, are of paramount importance for an arc of the parallel. There cannot be too many longitude stations in T consequence of that great stumbling-block in geodesy, the local deflections of the vertical or plumb-line. These deflections of the zenith from a normal direction have been divided into two groups: Those which are regional or manifest themselves with marked common features over thousands of square miles, and those which are quite local and greatly depend upon the surface features immediately surrounding them. These deflections, even in level countries, average about 2.5"; but in mountainous regions this deflection is greatly surpassed. Thus we find for deviation of the plumb-line at Patmos Head station 12" to the north, at Colorado Springs 25" to the west, at Salt Lake City about 17", and at Ogden about 15" to the east, at Genoa Station, Nev., nearly 29" to the west, the quantities depending to some extent on the spheroid of reference; but their amount and direction are obviously well accounted for by the position of the known attracting masses. In connection with this, continental attraction may manifest itself and be recognized by the astronomic amplitude of the longitudes of extreme stations of a long arc being in excess of the corresponding geodetic amplitude. The matter cannot be further pursued here in detail, but it may suffice to state that the average curvature of the equipotential surface of the geoid along the parallel of 39° approaches for about four-sevenths of the arc from its eastern A NOTE CONCERNING THE CHART ON THE OPPOSITE PAGE HE value of the Chart of the World, shown on the opposite page, is that the areas of all parts of the world appear in true proportion. The projection is the invention of Professor C. B. Mollweide, in 1805; numerous applications of it were made by Babinet in 1857, which gave rise to his name being attached to it under the designation Babinet's homolographic projection. It is an equal surface projection in which the entire surface of the earth is represented enclosed within an elliptic outline, whose major and minor axes represent the equator and central meridian respectively, with a ratio of 2 to 1. The parallels are straight lines, and the meridian, ellipses, and each zone or subdivision of the projection is in due proportion to the corresponding area on the sphere. The distances of the parallels from the equator-line are computed from the formula characteristic of the projection. C. A. S. end closely to that of the Clarke spheroid; whereas, for the remaining threesevenths, or for the region across the Rocky Mountains to the Pacific, the curvature comes more nearly to that of the Besselian spheroid. In the published paper two tables are given containing the results needed for combination with any other arc and, in conclusion, some preliminary rough combinations of American arcs are presented; all of which point to a reference spheroid of larger dimensions than those of the Besselian and are in favor of continuing the use of Clarke for reference. The second arc under consideration extends from Calais, Me., in the northeast and opposite the Canadian boundary, to the Gulf of Mexico, and terminates at New Orleans, La. It is known as the Eastern Oblique Arc of the United States. Its length is 2,612 km., or 1,623 statute miles; its difference of latitude is 15° 1', and of longitude 22° 47'. The general direction is, therefore, favorable and the length ample to secure fair results for an osculating spheroid. In the main the triangulation follows the Appalachian chain of mountains; in Western North Carolina and Eastern Tennessee it bifurcates, leaving an oval space between the two branches. The length of sides depends upon six base lines, and in general the development is closely accommodated to the hypsometric and other natural conditions along the course. It includes among its stations the two highest points in the eastern part of the United States, viz., Mount Washington, N. H., rising to about 1,920 metres, or 6,300 feet, and Mount Mitchell, N. C., rising to about 2,038 metres, or 6,687 feet. The adjustment of the whole triangulation is effected precisely as explained in the use of the arc of the parallel; the small reduction to the sea-level of the observed horizontal directions, on account of the altitudes sighted, was only applied when exceeding 0.05". The principal labor of adjustment was demanded by the necessity of bringing into accord the measured lengths of the Fire Island, the Massachusetts and the Epping base lines, and fulfilling the geometrical conditions of the intervening net of triangles. This demanded the satisfying of fiftyseven conditions and involved the simultaneous solution of an equal number of normal equations and the working out of one hundred and thirty-one corrections of observed directions. Of astronomic measures we have seventy-one latitude stations, seventeen longitude stations, and fifty-six azimuth stations, tolerably well distributed over the whole extent of the arc. The latitudes, as were those of the arc of parallel, were corrected for height of station or curvature of the vertical and for variation of pole according to Dr. Chandler's and Dr. Albrecht's researches. The same scrutiny as before had been extended to the deflections of the vertical, both regional and local. Partly on account of avoiding unnecessary labor, but principally on account of the crowding together of astronomic stations in certain very limited localities, and all of them, therefore, partaking of the deflections characteristic of this area, the total number of astronomic stations admitted into the final equations for the determination of the best spheroid were thirty-six for latitude, fourteen for longitude, and thirty-four for azimuth, or eighty-four conditions in all. These eighty-four differences between the astronomic and geodetic results constitute the data needed for a new determination of a spheroid; next the functional relations between the positions of these stations upon the reference spheroid to the earth's equatorial radius and to the compression of the polar axis had to be established. The final normal equations contain, therefore, four unknown quantities, viz.: the correction to the meridional deflection of the vertical at the initial or reference station of the oblique arc; second, the corrections to the deflection of the ver |