ble 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.

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.

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 | 299.15 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() fluctuated between the limits, and

285

The moon's path, on which the compression was not without influence, was also resorted to, and gave 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.†

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) executed 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′′ ·7 S.

(5.) The Peruvian arc, 1735-46, by Bouguer and la Condamine, from 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.‡

The writer does not explain why he has not taken in other reliable measures, in particular the Hanoverian by Gauss and the Danish are 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.

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, the French compared with the English gives, 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 measured arcs are—

the Russo-Scandinavian....

66 Indian

"French...

=25° 20′ 08′′.5,

=21 21 16 6,

= 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:

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.

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 "Tarqui

66

66

66

44° 23' 10" 3272650.1 "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

and its direction

3272671-5 toises

58° 44'

for the semi-minor axis of the equatorial ellipse 3272303-2 and its direction

Compression of the equator..

Polar compression of the greatest meridian

148° 44'

= 202'100

(( smallest 66

302.0041

and the separate arcs calculated with these values have the following deviations:

in toises.

in arc.

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 measured 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.

since it shows that each geodetic survey, and particularly the longitudinal differences, require correction. The writer shows from two examples how we must proceed and what we may expect.

Between Pulkowa and Dorpat the difference of longitude was found as follows:

-601 = 50.1 toises.

and the remaining error = 0"93 = Between Pulkowa and Warsaw is foundAstronomical 9° 17' 48"-43

9 18 01 24

+1281

7.8 toises.

124.7 toises.

Geodetic

Difference

The first error is thus completely got rid of and the second is reduced to nearly one half.

The writer concludes with the following remarks:

"This determination of the earth's figure is only an approximation, subject to many improvements, when we have more data, and when we use more rigorous methods of reduction. But it shows that we can obtain an agreement of the measures in which we have not heretofore succeeded. The determination of the general figure of the earth does not exclude local irregularities of its surface.

When this hypothesis of the earth's figure, after a scrutinizing investigation, finds general adoption, all geographical positions determined by geodetic means must have corrections applied, which will refer principally to the differences of longitude.'

As much as I may agree with this view and as certainly as the importance of the object requires the application of rigorous methods, it is yet clear that the trouble will be sufficiently rewarded when further reliable data, particularly from the western hemisphere, shall be available to the computer. It may be remarked here that the meridian in the western hemisphere corresponding to the small equatorial axes passes through Newfoundland, and, in the eastern, through the Amoor country and Eastern Siberia. For these countries we have at present not even approximate determinations. The Pennsylvania measure of 1764 is worthless for accurate investigation and the superior Peruvian measure is at too great a distance. Measurements of