EDGE OF THE EARTH'S SHAPE AND SIZE, BY THE UNITED STATES COAST AND GEODETIC SURVEY. By C. A. SCHOTT T HE Survey has just published a there were many years of interruptions. quarto volume containing an ac- What these long intervals signify and count of the transcontinental tri-. emphasize, is, that the various operaangulations and measurements of an arc tions of the Survey were more urgentof the parallel in latitude 39o. It also has ly required in the production of pracready for publication the manuscript giv- tical results for immediate use and in ing the result of an oblique are in the east- a great measure as aid to navigation. ern part of the United States. Both are For the more technical part of the work contributions of great length and among the available knowledge of the earth's the first of their kind in America. magnitude was sufficient for the early At first sight it might appear rather late needs. At the same time it was recogin the history of the Survey to bring out nized that the measurement of the earth results of the earth's figure. But it required the same means and methods should be remembered that such measures as that of an extended country, viz., were not the prime object of the Survey a net-work of primary triangulations and in its early stages of activity, but came a number of astronomical determinations about in the natural course of continuous for latitude, longitude, and azimuth of its development during nearly two-thirds of points. In time, therefore, sufficient maa century. What was required was to terial would accumulate to direct special secure a series of geodetic measures con- attention to this, the highest feature of sistent within themselves and serving as a geodesy. bond binding together the separate detail After the triangulations had reached surveys so as to form ultimately a sys- hundreds of miles in extent, and the geotematic whole. This requirement de graphical positions had been determined manded the establishment of extended by their development upon the surface of a primary triangulations not only along our spheroid representing the shape and size coasts, but also as a connecting link across of the earth, it became a matter of imthe country from ocean to ocean, to se- portance to see that the direct astronomcure uniformity of results. The growth ical measures for latitude and longitude of these operations depended of course kept in close accord with the correspondupon the immediate requirements of the ing geodetic measures; thus it came about Survey for the production of harbor and that in February, 1880, the Survey coast charts and was subject to the means changed its first reference spheroid, that available from year to year. of Bessel, for a more suitable one, that of Thus when I state that the first was Clarke of 1866. made between the years 1844 and 1898, When in 1889 the United States, by and the second between the years 1833 resolution of Congress, consented to beand 1898, I do not mean that it took fifty- come a member of the International Geofour and sixty-five years respectively to detic Association for the measurement of complete the task. Indeed, in either case the earth, the subject of the measures of arcs came into greater prominence, and consideration and the results reached may thus the field work of the two arcs, then now be briefly stated. First, the arc of fairly under way, was accelerated and the parallel in latitude 39°* It extends brought to a close late in the year 1898. from Cape May, N. J., on the Atlantic Before entering upon the detail of the coast, to Point Arena, Cal., on the Pacific two arcs it may not be out of place to coast, and ranges over 48° 46' of longistate that in order to obtain a measure of tude, with a linear development of about the dimensions of the earth, as repre- 4,225 kilometres, or 2,625 st. miles. The sented by a spheroid, that is, by a surface triangulation is supported by ten base 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 O. H. Tittmann, Superintendent, mathematical surface of equilibrium and U. S. Coast and Geodetic Survey. generally known as geoid. The definition of an osculating spheroid lines with an aggregate length of 532 thus implies that the sum of the squares st. miles, the longest or Yolo base being of the difference between the various as 10.9 miles in length; one-half of these tronomic and geodetic measures be a lines having a smaller probable error of minimum. The mathematical treatment measure than one part in a million. A of the combination of the arc measures differs according to their nature, whether *U. S. Coast and Geodetic Survey; H. S. they are extended in a certain direction or Pritchett, Superintendent. The Transcontiwhether large areas are covered, but in nental Triangulation and the American Arc of the Parallel. By C. A. Schott, Assistant, its generality it is necessarily laborious. Coast and Geodetic Survey, Washington, D. C., The salient points of the two arcs under 1900. characteristic of the triangulation is its consequence of that great stumbling-block rigidity imparted to it by quadrilaterals in geodesy, the local deflections of the verand other polygons. In crossing the tical or plumb-line. These deflections of Rocky Mountains many of its sides ex- the zenith from a normal direction have ceed one hundred miles in length, and been divided into two groups: Those there is one side reaching to a length of which are regional or manifest themselves 294 km., or 183 st. miles; the altitude with marked common features over thouof many of the stations is also consider- sands of square miles, and those which are able, reaching to 4,300 metres, or 14,108 quite local and greatly depend upon the feet, in the case of Pike's Peak, and to surface features immediately surrounding 14,421 feet at Mount Elbert. All geomet- them. rical conditions subsisting in the triangu- These deflections, even in level counlation are satisfied by adjustment, inclu- tries, average about 2.5"; but in mounsive of the required accord of the base tainous regions this deflection is greatly lines, so that the same length for any given surpassed. Thus we find for deviation of line is found no matter from what line the plumb-line at Patmos Head station one may start. This involved much heavy 12" to the north, at Colorado Springs 25 work; for instance, the triangulation ad- to the west, at Salt Lake City about 17", justment between the Salina and the El and at Ogden about 15" to the east, at Paso base demanded the simultaneous Genoa Station, Nev., nearly 29" to the solution of ninety-nine normal equations west, the quantities depending to some (with as many unknowns). In addition extent on the spheroid of reference; but the figures required the evolution of a their amount and direction are obviously correction to each of the two hundred and well accounted for by the position of the twenty-five observed directions. known attracting masses. In connection Coming to the astronomical measures, with this, continental attraction may manwe have distributed over or near the arc ifest itself and be recognized by the asone hundred and nine latitude stations, oc- tronomic amplitude of the longitudes of cupied almost exclusively with zenith tele- extreme stations of a long arc being in scopes; there are, also, seventy-three excess of the corresponding geodetic azimuth stations, various methods having amplitude. The matter cannot be further been used, and lastly we have twenty-nine pursued here in detail, but it may suffice telegraphically determined longitudes. to state that the average curvature of the These, of course, are of paramount im- equipotential surface of the geoid along portance for an arc of the parallel. There the parallel of 39° approaches for about cannot be too many longitude stations in 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 sphe- by the necessity of bringing into accord roid; whereas, for the remaining three- the measured lengths of the Fire Island, sevenths, or for the region across the the Massachusetts and the Epping base Rocky Mountains to the Pacific, the cur- lines, and fulfilling the geometrical condivature comes more nearly to that of the tions of the intervening net of triangles. Besselian spheroid. In the published This demanded the satisfying of fiftypaper two tables are given containing the seven conditions and involved the simulresults needed for combination with any taneous solution of an equal number of other arc and, in conclusion, some pre- normal equations and the working out of liminary rough combinations of Ameri- one hundred and thirty-one corrections can arcs are presented; all of which point of observed directions. Of astronomic to a reference spheroid of larger dimen- measures we have seventy-one latitude sions than those of the Besselian and are stations, seventeen longitude stations, and in favor of continuing the use of Clarke fifty-six azimuth stations, tolerably well for reference. distributed over the whole extent of the The second are under consideration arc. The latitudes, as were those of the extends from Calais, Me., in the north- arc of parallel, were corrected for height east and opposite the Canadian boundary, of station or curvature of the vertical and to the Gulf of Mexico, and terminates at for variation of pole according to Dr. New Orleans, La. It is known as the Chandler's and Dr. Albrecht's researches. Eastern Oblique Arc of the United States. The same scrutiny as before had been exIts length is 2,612 km., or 1,623 statute tended to the deflections of the vertical, miles; its difference of latitude is 15° 1', both regional and local. Partly on acand of longitude 22° 47'. The general count of avoiding unnecessary labor, but direction is, therefore, favorable and the principally on account of the crowding tolength ample to secure fair results for an gether of astronomic stations in certain osculating spheroid. In the main the tri- very limited localities, and all of them, angulation follows the Appalachian chain therefore, partaking of the deflections of mountains; in Western North Carolina characteristic of this area, the total numand Eastern Tennessee it bifurcates, leav- ber of astronomic stations admitted into ing an oval space between the two the final equations for the determination branches. The length of sides depends of the best spheroid were thirty-six for upon six base lines, and in general the de- latitude, fourteen for longitude, and thirvelopment is closely accommodated to the ty-four for azimuth, or eighty-four condihypsometric and other natural conditions tions in all. along the course. It includes among its These eighty-four differences between stations the two highest points in the east- the astronomic and geodetic results conern part of the United States, viz., Mount stitute the data needed for a new deterWashington, N. H., rising to about 1,920 mination of a spheroid; next the funcmetres, or 6,300 feet, and Mount Mitchell, tional relations between the positions of N. C., rising to about 2,038 metres, or these stations upon the reference spheroid 6,687 feet. to the earth's equatorial radius and to the The adjustment of the whole triangu- compression of the polar axis had to be lation is effected precisely as explained established. in the use of the arc of the parallel; The final normal equations contain, the small reduction to the sea-level of therefore, four unknown quantities, viz. : the observed horizontal directions, on ac- the correction to the meridional deflection count of the altitudes sighted, was only of the vertical at the initial or reference applied when exceeding 0.05". The prin- station of the oblique arc; second, the cipal labor of adjustment was demanded corrections to the deflection of the ver |