Oct, 3, 1878] 



NATURE 



603 



Scara}>ni 



lowed in calculating a portion of the English arc where pro- 

 ceeding from the Yorkshire coast between Redcar and Whitby 

 (the Euivey station is Easington) it crosses the sea in a line to 

 Saxaford in Shetland. At Easington astronomical observations 

 gave the azimuth of Cheviot as 38° 48' 58" •68. This determines 

 Sie meridian ; and the triangulation gives the relative distances 

 and azimuths at the following succession of stations (selected 

 for this calculation) : — Cheviot ; Mount Battock, on the Gram- 

 pians between Aberdeen and Balmoral ; Scarabin in Caithness, 

 Fitty Hill in the Orkneys ; Foula, a precipitous island between 

 Orkney and Shetland ; and Yell, a station in Shetland (not 

 quite so far north as Saxaford (see Fig. 3). 



On the meridian line defined above, take a point, A, whose 

 distance from Easington is equal to the side "Easington to 

 Cheviot." Join A with Cheviot and Mount Battock, and it is 

 evident that we can determine the distance A to Mount Battock 

 and the angle at A between Mount Battock and the north 



meridian. Next take the 

 point D at a distance from 

 A equal to the side " A to 

 Mount Battock." Join D 

 with Mount Battock and 

 Scarabin, then we can de- 

 termine the side " d to 

 Scarabin " and its inclina- 

 tion to the meridian at d. 

 Next take a point G (still 

 in the same meridian) at a 

 distance from D equal to 

 the side " D to Scarabin," 

 and join G with Scarabin, 

 Fitty Hill, and Foula. Next 

 take H at a distance from 

 G equal to the side " G to 

 Foula," and join H with 

 Foula, Yell, and Saxaford. 

 From Saxaford drop a per- 

 pendicular on the meridian 

 meeting it in s ; then we 

 have the distance Easington 

 to s, and the length of this 

 perpendicular, which is 

 about 200 feet. In order 

 to verify this result a dif- 

 ferent set of stations were 

 chosen for a second calcula- 

 tion conducted in a similar 

 manner ; the distance Eas- 

 ington to s by the first cal- 

 culation was 2288427*29 

 feet ; by the second it was 

 2288427-38 feet. 



The volume of the Ord- 

 nance Survey entitled "Ac- 

 count of the Prmcipal Tri- 

 angulation of Great Britain 

 and Ireland," by Capt. A. 

 R. Clarke, R.E., contains 

 (PP- 733-778) an elaborate 

 calculation of the figure of 

 the earth based upon the 

 English and French arcs of 

 eleven and twelve degrees 

 respectively, the Russian 

 and Indian of twenty-five 

 degrees and twenty-one re- 

 spectively, the first Indian arc, the Danish and the Prussian 

 arcs of about a degree and a half each, the Peruvian arc of three 

 degrees, and the Hanoverian arc of two degrees— a total length 

 of eighty degrees within a few minutes. The figure is first inves- 

 tigated without restricting it to the elliptic form, but assuming 

 that the radius of curvature of the meridian is expressed by the 

 formula p = A+ 2 b cos 2 4> -f 2 c cos 4 (^ (the ellipse is a particular 

 case of this curve, i.e., if 5 b^ - 6 A c = o). The resulting semi- 

 axes are a = 20927197 ft., b = 20855493 ft, a : /5 = 291 9 : 290-9, 

 and the meridian curve is more protuberant than an ellipse of 

 the same axes by the quantity 5 r = (177-5 ^ 709) sin'' 2 <p. But 

 when the cmve is restricted to the form of an ellipse the semi- 

 axes are found to be 20926348 and 20855233 with the ratio of 

 294-26 -. 293-26. The probable errors of the semi- axes so deter- 

 mined are ±186 and ± 239 respectively. 



Fig. 3. 



Subsequently to the publication ot the volume containing 

 these investigations, an extensive series of comparisons of 

 standards — the geodetic standaids— of length of the various 

 countries in which meridian arcs have been measured, was made 

 at the Ordnance Survey Office, Southampton. In order that 

 these most important comparisons might be made with the 

 utmost attainable accuracy, a building was erected especially for 

 this purpose. The comparison-room — itself double-walled — is 

 surrounded and "wholly inclosed within an outer building, and 

 it is partly sunk below the level of the ground. The diurnal 

 change of temperature is not perceptible in this room, conse- 

 quently comparisons are made under extremely favourable 

 circumstances. As a rule, each pair of standards was compared 

 in summer at about the temperatures of 62°, and in winter at 

 about 32° or 42°, but always at the natural temperature of the 

 room, artificial temperatures being wholly excluded. To give an 

 idea of the precision attained in these comparisons, the probable 

 error of the length of the Ordnance Standard ten-feet bar in 

 terms of the national yard is four ten-millionths of a yard ; that 

 of the Indian steel standard (a ten-feet bar) is less than three 

 ten-millionths of a yard ; that of the Russian standard, a bar 

 of more than twelve feet long (two toises) is six ten-millionths. 

 The importance of these results, connecting the largest arcs, upon 

 which the figure of the earth must depend, can hardly be over- 

 estimated. The length of the "Toise of Peru" obtained 

 through three entirely independent sources, viz., the Russian, 

 the Prussian, and the Belgian toises, is 6-39453348 feet, from 

 which the greatest divergence of the three separate results is only 

 half a millionth of a toise ; this corresponds to ten feet in the 

 earth's radius. The length of the " metre " deduced from the 

 above by means of its defining ratio (443296 : 864000) is 

 3-28086933 feet. 



One of the most difficult of the determinations of length made 

 in connection with this series was the measuring the distance 

 between the knife edges of Kater's Reversible Pendulum. In 

 this operation the flexibility of the bar was in the preliminary 

 comparisons a source of much error. This, however, with some 

 other troublesome matters, was successfully overcome, and the 

 length was determined with a probable error of less than one 

 five-millionth part of the length. The length of the pendulum 

 was obtained by comparing it with the metre, and the details of 

 the measurements will be published in the "Account of Pendulum 

 Observations in India," by CoL Walker, R.E., F.R.S., Superin- 

 tendent of the Great Trigonometrical Survey and Surveyor- 

 General of India. 



These results of the comparisons of standards of the dif- 

 ferent countries concerned, modify the numbers given above 

 for the semi-axes of the earth. After making the necessary 

 corrections, the semi-axes of the elliptic meridian are 20926062 

 feet and 23855121 feet, or 6378206-4 metres and 63565838 

 metres. The ratio of the axes 293-98 : 294-98. See the 

 Ordnance Survey volume, entitled " Comparisons of Standards 

 of Length," page 287. In the same volume will be found an 

 investigation of the figure of the earth, supposing it to be an 

 ellipsoid of three unequal axes. 



But these results are now again superseded by a more recent 

 determination published in the Philosophical Magazine for 

 August. It appears that during the last few years the Surveyor- 

 General of India, Col. Walker, C.B., F.R.S., Royal Engineers, 

 has been not only measuring new arcs in India, both of latitude 

 and of longitude, but has revised the southern portion of the 

 Indian arc as measured originally by Col. Lambton. On this 

 chain of triangles considerable doubt rested as to what was the 

 unit of length used in the measure ; a complete remeasurement 

 according to modern methods has set this question at rest. A 

 complete meridian chain of triangles has been carried from 

 Mangalore on the west coast in latitude 12° 52' and longitude 

 75° E., to a point in latitude 32° ; thus the whole Indian arc is 

 now 24° in length. Eleven differences of longitude have been 

 determined by electro-telegraphy between the stations Mangalore, 

 Bombay, Vizagapatam, Madras, Bangalore, Hydrabad, and 

 Beliary ; it is almost unnecessary to add that in these operations 

 no refinements of modem science have been overlooked. As 

 the difference of longitudes of Bombay and Vizagapatam is 

 10° 28' and the geodetic connections of the above seven stations 

 is completed (liable, however, to some small future corrections 

 resulting from the least square calculations) there is thus pre- 

 sented a large addition to the data of the problem of the figure 

 of the earth. Taking the English and French conjoined arc of 

 22° with fifteen astronomical stations ; the Russian arc of 25° 

 with its thirteen stations ; the Indian meridian arc of 24° with 



