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ART. VI.—The True Figure of the Earth.-Notice by Mädler in

Prof. Heis' Wochenschrift für Astronomie, Meteorologie und Geographie, No. 51 and 52. Dec. 21 and 28, 1859.*

Essai d'une détermination de la véritable figure de la terro. Par T. F. DE ScHoBERT, St. Petersbourg, 1859.

This brief but very important paper treats of a question which has engaged mankind for ages, and treats it in such a way as to convince us that an essential step forward has been taken towards its final determination.

It is not intended to recapitulate the history of opinions and notions which antiquity, as well as the middle ages, had formed on this topic so generally interesting, as it does not belong to the plan of the work to be discussed.

Since the establishment of Newton's theory, that the earth is flattened at its poles, and the confirmation of this theoretical result by the first measurements of arcs in the eighteenth century, the chief inquiry has been directed to the magnitude of this compression, or, in other words, the difference between the semi. major and the semi-minor axes of the earth. The attempt was made to determine it by comparing arcs measured in different latitudes, the latitudes of their extremities being astronomically and their extent on the earth geodetically determined. In this manner was obtained, between 1735 and 1746, two arcs, the one in Peru, and the other in Lapland, which could be compared with each other, as well as that previously measured by Cassini in France. The result of the comparison was not satisfactory. Although they proved a comparison, the measures did not agree, and the source of this difference was too obscure to favor the supposition that any nearer approach had been made to the ob. ject of research.

In the course of the eighteenth century measures of arcs of the meridian were executed or attempted in France, Austria, Italy and Pennsylvania, at the Cape of Good Hope and other places. Their comparison made it apparent that the compression G47), as at first concluded, was too great, and that it must be reduced one half or even more. Yet the uncertainty remained very considerable.

The measurement of arcs was continued in the nineteenth century over greater extents in different parts of the earth and

with greater care and accuracy, and the close agreement of these . arcs, as it appears from the comparison of their parts, left but little, further to be desired. Perceptibly different values for the compression were nevertheless obtained, as when the most probable value was determined from them all, there still remained from the separate measures of arcs, deviations which could not be ascribed to errors of observation.

* Translated for the American Journal of Science by CHARLES A. Sorort, Assistant, and communicated by Prof. A. D. BACHE, Supt. U.S. Coast Survey.

Repeated investigation has shown that the old measurement of a degree by Maupertuis and Outhier was considerably in error. Rosenberg and Swanberg obtained quite different values. In like manner the measures executed by Boscovich, Le Maire, Liesganig, Beccaria and others in the eighteenth century could not bear severe criticism nor stand side by side with the better and approved arcs. A more accurate and reliable coefficient of compression still remains a desideratum.

We insert here for comparison the values of the compression, as they result, for the most part, from the same measures, but according to different methods of reduction, in the first four decades of this century.

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Semi-major axis | Semi-minor axis Compression Coeff}. of
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compression. | 3271742:8 3260989.3 / 10785.5

| 302.78 3271852.3 3260853.7

10998.6 797.48 3271953:9 | 3261072.91 10881.0 l god.7)

As Bessel afterwards discovered an error in Delambre's French triangulation which could not be without influence on the combined result, he repeated his reduction with this correction and found

Bessel, (II), - | 3272077.1 | 3261139.3 | 10937.8 | 295:15 Still, according to the last reduction, the probable error is three times as great as the difference of Bessel (I) and Bessel (II).

Since in this even the best measures of this sort do not yield sufficiently accordant results, it was attempted to find the compression in two other ways. The observations of the pendulum give, as a consequence of the earth's compression, different lengths of the seconds pendulum in different latitudes, and hence we can deduce, conversely, the compression from these observed lengths or, if we choose, from the number of beats which the same pendulum makes in the course of a day. Observations made with extreme care were used as the foundation of such experiments, yet still the result (zob.zo) fluctuated between the limits, is and his

The moon's path, on which the compression was not without influence, was also resorted to, and gave göz for the compression, but on account of the smallness of the aggregate effect it is of inferior accuracy. *

We have found it necessary to present this general view in order to define the limit of the results arrived at by preceding

* This result depends, in a measure, on the law of density of the earth in passing from the surface to the centre.

efforts, and in order to place the merit of the writer in its proper light, especially for those not possessing a special knowledge of the subject in question.

The preceding investigations were all based upon the following assumptions:

(1.) The meridians of the earth are ellipses.*
(2.) The minor axis is also the axis of rotation.
(3.) All the meridians of the earth are equal.

The writer remarks that, in a rigorous sense, no one of these assumptions is proved and that we do not possess the means of proving the first two. We may however add, that these two assumptions, if not absolutely, must be very nearly true.t

In regard to the third assumption we are now prepared to submit it to investigation, and the previous failures compel us to question its applicability. Paucker and Borenius have already attempted to prove that it must be false, but neither has arrived at any definite result. The writer uses as a basis the following measurements of arcs:

(1.) The Russian (or more properly Russo-Scandinavian) exe. cuted in 1820-51 by Hansteen, Selander, Struve and Tenner from Fuglenäs (in latitude 70° 40' 11" :3 N) to Staro-Nekrasofka (45° 20' 02" .8 N) the longest arc yet measured.

(2.) The Indian arc, 1802-43, by Lambton and Everest, from Kaliana (29° 30' 48" 9 N) to Punnæ (8° 09' 32" :3 N).

(3.) The French arc, 1792–1806, by Méchain, Delambre, Biot and Arago, from Dunkirk (51° 02' 08" 5 N) to Formentera (38° 39' 56' •1 N). . (4.) The measure at the Cape of Good Hope, by Henderson and Maclear, from 34° 21' 06" :3 S to 29° 44' 17" S.

(5.) The Peruvian arc, 1735–46, by Bouguer and la Condamine, fron Tarqui (3° 04' 32" 1 S) to Cotchesqui (0° 02' 31" 4 N).

(6.) The Prussian arc, 1831-34, by Bessel and Bæyer, from Memel (55° 43' 40" 4 N) to Trunz (54° 13' 11" •5 N).

(7.) The British arc, by Roy and Mudge, from Clifton (53° 27' 31" :1 N) to Dunnose (50° 37' 07" :6 N).

(8.) The Pennsylvanian arc, 1764, by Mason and Dixon, from 39° 56' 22" •5 N to 38° 27' 37" •5 N.1

The writer does not explain why he has not taken in other reliable measures, in particular the Hanoverian by Gauss and the Danish arc by Schumacher; but his principal aim being to prove merely the untenability of the assumption that all meridians are equal and similar, and pronouncing his own work as preliminary, no exception can be taken to this course. In a perfectly rigorous investigation by the method of least squares, where each

* In the recent account of the Ordnance Survey of Great Britain the figure of the meridian, in one of the hypotheses, is not restricted to this condition. + The truth of the second assumption may be granted.

The reason for taking in this arc will be apparent from what follows.

greater is, and th:7, the frethe

measure with its corresponding weight would enter, such omissions could of course not be permitted.

Reducing and comparing with each other, each of the eight arcs, he obtains twenty eight binary combinations, which he communicates in detail; but this can only be for the purpose of showing the impossibility of an agreement by the assumption hitherto made. For instance, the Russian arc compared with the English gives 5:7, the French compared with the English gives 76.5, and the deviations from the general mean are still greater in comparing the Prussian and Pennsylvanian arcs with all the others. It is thence inferred that the meridians of the earth are not equal to each other, and thence that the equator and the parallels are not circles, so that it is generally impossible to draw a great circle on the earth's surface anywhere.

Meridians different in form and length really indicate different polar compressions, a non-agreement of the results found, proving nothing against the accuracy of the measures.

Now since all meridians must converge at the poles they must all have in common one and the same diameter, viz., the smallest (the axis of rotation) which can be obtained from each of the large arcs referred to. The three greatest ineasured arcs are

the Russo-Scandinavian............ = 25° 20' 08":5,
“ Indian ............... .. = 21 21 16 .6,

“ French........ .......= 12 22 12 •4, whilst all the others, including those not used by the writer, are less than 5° in extent.

By dividing each of these three arcs into two equal parts and comparing one part with the other and also each part with the whole, the writer obtains the following mean values:

Semi-major axis | Semi-minoraxis
From the Russian.....


31 3261428.7 " " Indian............. 3272650.

9 1 3261547.4 66 " French ............ 3273448.2 1 3260364.7 In the first two, the differences are unimportant, but the last shows a greater deviation. Schubert remarks that this latter deviation can be got rid of by changing the latitude of Carcassonne, the selected point of division of the French arc, by 1":96, which may not exceed the limits of uncertainty. He considers it safer, however, to obtain the semi-minor axis from the first two arcs alone, the greater arc (the Russian) having twice the weight of that of India. In this manner he deduces the semiminor axis :

3261467.9 toises. SECOND SERIES, VOL. XXX, No. 88.-JULY, 1860.



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By means of this value the semi-major axis can be found for each measure of an arc. It results as follows: For the meridian of Kaliana in Long. 95° 20' | 3272581.3

“ Dorpat " " 44° 23' 10" | 3272650:1

* “ Tarqui “ “ 298° 44' 3272382-8 Three radii and their included angles suffice for the determination of the ellipse. The writer finds for the semi-major axis of the equatorial ellipse

3272671-5 toises and its direction

58° 44' for the semi-minor axis of the equatorial ellipse 3272303.2 and its direction

148° 44' Compression of the equator.............. = Joroi Polar compression of the greatest meridian = 500Too 16 "

" smallest " = Jozioni, and the separate arcs calculated with these values have the fol. lowing deviations :

in toises.

in arc. Peru,........

+ 1.2

0":077 Pennsylvania,..

- 105.68

6 .687 England, ..........

+ 11.68

0736 France, ...........

- 25:45

1 .607 Cape of Good Hope,.

- 6.98

0.442 Prussia,.......

+ 23.28

1 .267 Russia, ...........

- 20:47

1 .289 India,. ...... .......+ 25.54

1 .619 In reference to the arcs showing greater deviations the writer remarks that the Pennsylvanian arc was measured with imperfect means (by the chain only, and without triangulation) and that he has taken it merely for trial.* The other seven deviations are so small that no one exceeds the ten thousandth part of the meas. ured length, and the great accnracy of the Peruvian measure, which had been already shown in former discussions, appears here in its true excellence. Honor is justly due to the memory of the men who, at so early a period, accomplished such a work under the untold difficulties of ten years' separation from all civilized countries.

It has frequently been shown that the geodetic measures, in particular the longitudes, do not harmonize with astronomical observations, but no one has yet succeeded in discovering the cause of this disagreement. The ellipticity of the earth's equator, as discovered by the writer, will call forth new investigations,

* Dr. Maskelyne in his description of Mason and Dixon's base (Phil. Trans., London, 1768) says they employed rods of fir frequently compared with a standard brass measure at a fixed temperature. This line was revised in 1849 and 1850 by Lt. Col. Graham, U.S. Top. Eng. See Message of the Gov. of Md., &c., in relation to the intersection of the boundary lines of the States of Maryland, Pennsyl. vania and Delaware. Washington, 1850.

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