Our Knowledge of the Earth's Shape and Size 37 



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 repre- 

 sented 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 ob- 

 tain 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 sur- 

 face an osculating spheroid may be de- 

 termined, 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 as- 

 tronomic 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 



consideration and the results reached may 

 now be briefly stated. First, the arc of 

 the parallel in latitude 39°.* It extends 

 from Cape May, N. J., on the Atlantic 

 coast, to Point Arena, Cal., on the Pacific 

 coast, and ranges over 48° 46' of longi- 

 tude, with a linear development of about 

 4,225 kilometres, or 2,625 st. miles. The 

 triangulation is supported by ten base 



O. H. Tittmann, Superintendent, 

 U. S. Coast and Geodecic Survey, 



lines with an aggregate length of 53^ 

 St. miles, the longest or Yolo base being 

 10.9 miles in length; one-half of these 

 lines having a smaller probable error of 

 measure than one part in a million. A 



* U. S. Coast and Geodetic Survey; H. S. 

 Pritchett, Superintendent. The Transconti- 

 nental Triangulation and the American Arc 

 of the Parallel, By C. A. Schott, Assistant, 

 Coast and Geodetic Survey, Washington, D. C, 

 1900. 



