(only a little over one-nine-thousandth,) and the actual difference between the major and the minor axis is less than one mile. The difference is sufficient, however, to invalidate the assumption of the scientific commission of 1799, under whose advice the basis of the metric system was

Hitherto, it has been generally held by geodesists, (1.) that the meridians of the earth are ellipses; (2) that the axis of rotation is the minor axis of all these elliptical meridians; and (3,) that the meridians of the earth are all equal. On this hypothesis, it would not matter in how widely different longitudes different degrees of latitude should be measured; when compared, they ought always to give the same values for the polar and the equatorial diameters of the spheroid, and for the compression of the poles. The fact is, that the results deduced from such comparisons are largely discordant.

Gen. de Schubert's paper commences with a series of comparisons of this nature. For the purposes of the comparison he selects eight different arcs, viz: the great Russian arc, of 25° 20′ in length; the Indian arc, 21° 21' the French arc, (extended to Formentera,) 12° 22'; the South African arc, (of Maclear and Henderson,) 4° 37'; the Peruvian arc, 3° 7' ; the British arc, 2° 50′, (since extended to 10° 13′;) the Prussian arc, 1° 30′; and the Pennsylvania arc, 1° 29′. Five of these eight arcs differ much less in longitude than could be desired, being all within a range of less than thirty degrees. The Indian and the Peruvian arcs differ in longitude by nearly half a circle, but the Peruvian is very short. The Pennsylvanian arc is nearly in the same longitude as the Peruvian, and seems to have been included, not as having important weight, but because of this circumstance.

Comparing each of these eight arcs with every other, the author obtains twenty-eight systems of elements, presenting great discordances. The maximum and minimum values obtained for the semiaxes differ by miles, and the values found for the compression are equally various. Gen. de Schubert concludes from this that the earth cannot be a solid of revolution; but he still holds that the meridians are elliptical; and he consequently infers that the true mode of finding the figure of the earth is to compare different portions of each arc with other portions of the same arc, or with the whole.

When this conclusion is reached, however, we see at once how meagre are the materials available for the application of this method. The Russian arc a little exceeds one-fourth of a quadrant in length, and the Indian arc falls short of one-fourth of a quadrant by about the same amount. The French arc (extended to Formentera) is about one-seventh of a quadrant. These are long enough to permit of some comparisons, tolerably trustworthy, to be made within themselves; but the rest in the list above given may for this purpose be dismissed at once.

Now dividing these three principal arcs into two portions approximately equal, each, the author obtains from the Indian and the Russian, values of the polar axis differing only about fifteen hundred feet; but the difference between the values of this element as deduced from the Indian and the French arcs, is more than ten times as great, or exceeds fifteen thousand feet.

The author therefore rejects the French arc in making this determination, thus narrowing his base to the Indian and the Russian alone; giving at the same time, rather arbitrarily, double the weight to the Russian which he gives to the Indian.

By the aid of the semiaxis thus found, and the measured length of the degree of Peru, the equatorial radius in the longitude of the Peruvian arc is obtained; and this, with the Indian and the Russian equatorial radii, serves to determine the eccentricity of the equator, considered as

selected; the assumption, viz: that the earth is a regular oblate spheroid, all the sections of which, made by planes passing through the axis of revolution, are equal and similar. On this assumption, the ten-millionth part of any one meridional quadrant is the ten-millionth part of any other; and wherever a man may be upon the surface of the planet, he has beneath his feet the natural standard upon which rests the system of

an ellipse, and the position of its major and minor axes. To find the length of a meridional quadrant in any longitude, the next step is to calculate (which with the data now possessed is easy) the length of the equatorial radius in that longitude. This is the major semiaxis of the meridional ellipse, and the earth's polar semiaxis is the minor.

The idea of this method is excellent, but it rests on assumptions which are only approximately true, and it requires that more numerous and more extended measurements should be made before it can be satisfactorily applied. It assumes that the meridians are all elliptical, but none of them appear to be strictly so. It assumes the equator to be an ellipse, but the equatorial diameters independently deduced from the several meridional arcs, do not well sustain that hypothesis. Capt. Piazzi Smyth, Astronomer Royal of Scotland, expresses the opinion (“Our Inheritance in the Great Pyramid,” p. 38,) that they prove it rather to be “an irregular curvilinear triangle." There is, furthermore, a mechanical difficulty involved in Gen. de Schubert's theory of the earth's figure; which is, to explain how a planet of which the surface is three-fourths, and the equatorial circumference nearly five-sixths, fluid, should have the form of an ellipsoid of three unequal axes.

It was suggested by Prof. Airy, as above stated, on examining the results of Gen. de Schubert, that a better mode of employing the available material would be to make no attempt to determine in advance the value of the earth's polar axis, or any of its equatorial radii, but to leave the three semiaxes of the ellipsoid, as well as the longitude of the equa torial semiaxes, indefinite; and to determine, by the method of least squares, what values given to these would best represent all the posi tions of all the stations which had been determined astronomically and geodetically upon the several arcs measured. This was the method employed by Capt. Clarke, in his elaborate investigation presented to the Royal Astronomical Society in 1860, and published in that year in the 29th volume of their Memoirs. Capt. Clarke selects forty stations upon the lines of the Indian, the Russian, the French, the British and the Peruvian arcs, and determines what are the values of the variable elements mentioned above, which make the squares of the errors of latitude of these stations a minimum. He thus finds a larger eccentricity in the equator than Gen. de Schubert, and a smaller polar axis; also a larger eccentricity of the Paris meridian, and a larger error of the metre. Capt. Clarke has several times modified his results, as reason has been found to correct the latitudes of some of his stations; and he appears to be by no means satisfied that the equator is truly an ellipse and not a circle. On this point his own langnage is: “ "Whatever the real figure of the earth may be, if on the investigation we presuppose it to be an ellipsoid, it is quite clear that the arithmetical process must bring it out an ellipsoid of some kind or other; which ellipsoid will agree better with all the observed latitudes, as a whole, than any spheroid of revolution will. Nevertheless, it would scarcely, I conceive, be correct to say we had proved the earth not to be a solid of revolution. To prove this would require data which we are not in possession of at present, which must include several arcs of longitude. In the mean time, it is interesting to ascertain what ellipsoid does actually best represent the existing measurements."

weights and measures for the world. But since this assumption has been shown to be possibly or probably incorrect, we are no longer at liberty to regard the ten-millionth part of a quadrant of a meridian as being a quantity everywhere the same. A metre deduced from the great meridional arc of Russia would be slightly greater than one derived from the arc of Peru. The actual metre, supposing it to be truly the ten-millionth part of the French quadrant, would fall very nearly half way between these values; since, according to de Schubert, the radius of the equatorial ellipse lying in the plane of the French arc is very nearly the mean equatorial radius; while the similar radii corresponding to the Russian and Peruvian meridians, are not far from the positions of the equatorial semiaxes.* This discovery, if it is proper to apply such a term to what is as yet but a plausible hypothesis, renders it necessary to qualify the definition of the metre, and to say that this unit is the ten-millionth part, not of a quadrant of any terrestrial meridian, but of a quadrant of a particular terrestrial meridian. Whatever there may be, therefore, pleasant to the imagination in the idea of a standard derived directly from a dimension every where the same, and every where equally ascertainable, of the globe on which we live, must be relinquished. This consideration would no doubt have been fatal, in the view of the scientific commission of 1799, to the claims of the meridian as a basis of a system of weights and measures, had the irregularity of the earth's figure been known at the time it was selected for that purpose; for the commission rejected the proposition to adopt the pendulum as the basis, for the reason of a similar want of uniformity of the indications of such a standard in different latitudes and different longitudes. It must be obvious, however, that it is only the ideal perfection of the standard, scientifically considered, that is impaired by the discovery of the irregularity of the terrestrial ellipsoid. Practically this circumstance is of no importance whatever. If the standard had been some natural dimension to which reference could upon occasion be easily made, either directly by individuals, or by the combined efforts of several, exacting in practice no great labor or expenditure of time or of money, then the discovery that this dimension, originally assumed to be everywhere the same, is not so, would be one of gravity. But the quadrant of the meridian has no such universal availability as this. It was not contemplated by the authors of the metric system that the stupendous labor of measuring a great meridian arc would be ever again undertaken for the purpose of simply verifying the length of the metre, or of recov

*The positions of these axes are, however, very imperfectly ascertained, if indeed the whole hypothesis of the ellipticity of the equator is not a mistaken one. Capt. Clarke's paper, referred to previously, removes the vertex of the equatorial ellipse from longitude 41° East to longitude 14° East. This would give to the Paris meridian nearly a maximum instead of a mean length. In his revised calculation, published in 1866, in an appendix to a volume from the Ordnance Survey, containing Comparisons of Standards of Length of England, France, Belgium, Prussia, Russia, India and Australia," he removes it again from 14° East to 15° 34' East.

ering it, in case of its accidental loss. The latter accident was provided for, by directing that the metre should be re-established, in case of the destruction of the prototype deposited at the palace of the Archives, by means of its known relation in length to the length of the pendulum vibrating seconds at Paris. But it is not probable that even this comparatively expeditious method would be practically resorted to, since pendulum experiments of this degree of delicacy are difficult; and are themselves subject to some uncertainty. The probability rather is, that the prototype metre would be replaced, if lost, in the same manner in which the standard British yard was reproduced after its destruction by the burning of the Parliament houses in 1834; and that is, by the comparison of copies of it carefully made previously to its destruction, of which considerable numbers now exist. The British statute on the sub

ject, like the French, required the reproduction to be made by reference to the seconds pendulum; but since the renewal of the standard, this provision of law has ceased to exist in Great Britain.

Looking at the matter practically, therefore, it may be stated that the metre is the length of a certain platinum bar, originally constructed of the exact length, as presumed, of one-ten-millionth part of the terrestrial meridional quadrant passing through Paris; this length having been determined by an elaborate measurement of nearly one-ninth part (9° 40′) of that quadrant. The paramount reason for the selection of such a standard originally, was that the unit might be as invariable as the globe itself. This property of invariability it has, in being derived from a particular meridian, quite as completely as it could have it if all the meridians were equal. The fact that the metre represents the ten-millionth of one particular quadrant, is only to be regretted, inasmuch as it detracts from the beauty of the pure ideal upon which the system was founded.

But it is objected that the actual metre is not, after all, the exact tenmillionth part even of this particular quadrant. It is said to be too small by a fraction, minute indeed, but by no means inappreciable. This objection, which is apparently not without foundation, seems by some to be regarded as a sufficient reason for rejecting the metre as the basis of a system of weights and measures altogether. Such persons, to be logical, should reject equally every basis which purports to be a determinate part of any given dimension of the earth, whether it be of a meridian, or of the equatorial circumference, or of the polar axis, or of the equatorial or the mean diameter, unless this dimension shall be, or until it shall be, demonstrated to have been ascertained with absolute exactness. But that certainly is not the case at present with any of the dimensions just named. If anything is made apparent by an examination of the details of any of the great geodetic operations which have been carried on during the last two hundred years, it is that the earth is too irregular in figure to be regarded any more as an ellipsoid of three axes, as Gen. de Schubert and Capt. Clarke would make it, than as a simple oblate spheroid; and yet these gentlemen profess to assign the error of the metre, by comparing the quadrant passing through Paris, theoretically computed as a quadrant of

such an ellipsoid, with the same quadrant as determined by the actual measurement of its ninth part. But certainly this arc measured did not form part of a regular ellipse. Had that been true, the successive degrees measured would have exhibited, in proceeding from south to north, a regular and gradual increase in length. An increase was observed, but it was by no means regular. The whole arc being divided into three parts approximately equal, showed, in the southern division, a mean increase of 12.9 toises; in the middle division, an increase of 324 toises; and in the northern division, an increase of 5.5 toises only. When subsequently the arc had been extended northward to the latitude of Greenwich, and southward to the island of Formentera, making a total of 12° 48′ 43′′.89, or very nearly one-seventh of the quadrant, a similar division into five parts, (the northern, however, being less than the average of the others,) gave, in the southern division, a diminution going north, in direct contradiction to the theory of a flattened spheroid. The succession of values was then, for the southernmost division, a diminution of two toises per degree going north; for the second, an increase of 12.9 toises; for the third, an increase of 32-4 toises; for the fourth, an increase of 84 toises; and for the fifth, an increase of 7.23 toises. It is thus manifest that the diminution of curvature was least rapid in the middle of the arc, being more rapid both north and south of that point, which is not a characteristic of an elliptical curve. The scientific commission which fixed the length of the metre, had before it only the measurement from Mont Jouy to Dunkirk, of 9° 40′ 45′′.67. Two eminent geometers, members of this commission, Laplace and Legendre, found, from a comparison of the different sections of this arc, an ellipticity of 1-150th and 1-148th; but from a comparison of the whole with the arc of Peru, only an ellipticity of 1-334th. Delambre, who, with Méchain, had executed the measurement, found, from a similar comparison with the Peruvian arc, an eccentricity of 1-312th, and subsequently of 1-309th. Upon the extended arc, from Formentera to Greenwich, he made the eccentricity finally 1-178th. The commission fixed the metre at 443-296 French lines; but Delambre concluded, from his latest results, that the true length should be 443-320 lines; which, if correct, would show the metre of the commission to be too short by 1-460th of an English inch.

Degrees in the same latitudes, measured in different parts of the world, differ, in some instances, very sensibly. The length of the degree found by Liesganig, in Hungary, is materially less than that in the corresponding latitude in France. That found by Mason and Dixon, in Pennsylvania, is also considerably less than that of the French geodesists between Formentera and Mont Jouy, both in latitude 39°. A very extraordinary anomaly is presented by the measurement of Lacaille, at the Cape of Good Hope, which gives, in latitude 33° S., a greater length than is found in France in latitude 45°. This determination, compared with the Peruvian arc, would imply, in the southern hemisphere, the extreme flattening of 1-78th.*

* Little weight is attached to the measurement in Pennsylvania, it