deciding whether the curvature of the two terrestrial hemispheres should be regarded as identical or distinct. At the end of the last century, as has been before intimated, La Caille had transported himself from France to the Cape of Good Hope, in the design of co-operating in the solution of various astronomical problems which in that remote country seemed to call for an intelligent observer, and having there executed the measurement of a small arc of the meridian, he obtained for the irregularity of the globe a much smaller value than any of the analogous ones deduced in Europe and America. How was this anomalous result to be explained? In one of two manners: either by attributing it to a real defect of symmetry in the form of the earth, or to an error, not easily to be avoided, in the operations of La Caille; but as the first was contradictory of the received theory and opposed to many facts well ascertained by other observers, and as the second was scarcely admissible in view of the recognized talent, industry and conscientiousness of the French savant, no one knew which alternative to adopt. Everest, on his return from India, inspected the locality where La Caille had operated, and at sight of the mountains which surround it concluded that the distinguished astronomer might easily have deceived himself, or neglected certain precautions without which no geodesic labor can really afford a sufficient guarantee of certainty. Maclear, with due regard to the indications of Everest, undertook in 1837 an operation analogous to that previously executed by La Caille, though on a larger scale and with better material resources; and the result now confirmed the previsions of the theory, or the identity of form of both terrestrial hemispheres.

Nor has it been only in a direction from north to south that astronomers and geometers have essayed to estimate the dimensions of the earth. When a comparison of the first results obtained in the proceedings directed to that object had revealed, not only the defective sphericity of our globe, but the irregularities or accidents which interrupt its presumed ellipticity, whether from one pole to the opposite, or even in passing from one meridian to another not far distant, the attempt was also made to measure one or more arcs of parallel. That by this means, as by the former, and still better by a combination of both, a knowledge of the form and volume of the earth might be obtained, is readily conceived; and when it is considered that this new operation is even more delicate and troublesome than the other, the reader will scarcely wonder that till a quite recent epoch the number of arcs of parallel measured bore no proportion to the arcs of meridian. The Franco-Spanish commission, charged with measuring an arc of the latter sort in Peru, proposed also to determine the value or a degree of parallel, which in those regions would have been a degree of the equator, but a difference of views as to the execution, added to the difficulties of the enterprise, led to a relinquishment of the project before it had begun to be carried into effect. At the same epoch, 1734 to 1740, the Cassinis, Maraldi and La Caille measured in France two arcs of parallel, one in the latitude of Paris, the other across Provence; and still later Lambton undertook in India a work of the same kind; but these first essays led to no definite result, and served only to show at once the utility of the undertaking and the difficulties which its adequate accomplishment would present. The measurement of a great are of parallel, stretching from the neighborhood of Bordeaux to Padua, or from the ocean to the Adriatic, over an extent of 13° and at a latitude of 45° 43', was commenced in 1811 under the direction of Colonel Brosseau, and continued in 1820 across upper Italy by Carlini, Plana and other astronomers and geometers of Italy, France and Switzerland. Besides this operation, which forms an epoch in the annals of geodesy, there must also be mentioned the measurement of another arc upon the parallel of Paris, from Brest to Strasburg, executed between 1818 and 1823, by the French functionaries Bonne and Henry; another completed from Greenwich to Valentia in the west of Ireland, by Professor Airy; and a third commenced in 1857 by W. Struve, in 52° of

latitude, designed to connect with the last, and thus embrace an arc of about 70° total length, from one extremity of Europe to the other.

It would be impossible, without overstepping the limits which discretion prescribes, to carry further this enumeration of geodesic labors already accomplished, or in course of execution, or projected for early realization. The earth would be seen to be covered with an immense net-work of triangles, whose meshes interlace more and more every day, so as to leave to truth thus earnestly sought less and less chance of finally evading detection. In this work of so many ages, where, more perhaps than in any other, man has displayed the talent and irresistible energy with which he is endowed, Spain is to-day taking an active and honorable part. The summits of our mountains, although constantly visited by distinguished military functionaries, resound not now with the echoes nor are seen clothed with the smoke of battle. They serve not as watch-towers of war, but as stations for geodesic signals, true symbols of peace and of culture.

But, as is opportunely asked, in order to dispel the doubt, by one of the most estimable intellects of our country,* To what end do so many measurements of the globe conduce? What practical result is expected from such laborious and persevering attempts? Of results to be appreciated by the material and tangible interests involved, perhaps none; but does science propose for its exclusive object the satisfaction of man's primary necessities? Hunger and thirst appeased, is there, indeed, nothing beyond? Wretched would be the science which would shut itself up within such narrow limits, which should restrict the soul to the care of its frail tenement, and seek in the secrets of nature no trace of its Creator, which should refuse to lift itself from the abject to the elevated, from the slough of earth to the etherial regions of infinity. And taking for granted that there is nothing fortuitous in the universe, and that the earth, instead of being spherical, is elliptical, or of a more complicated form, does not science fulfil its appropriate task when it investigates the true figure of this little globe of ours, not for the simple pleasure of knowing it, but with the further purpose of discussing the reason of that form, its origin, the changes experienced, the perturbations by which it may have been affected, the influence it exerts or the function it fulfils in the admirable co-ordination of the created whole? If all this is not worth the trouble of investigation, to what other mystery of the physi cal world should man, in preference, consecrate his studies?

III.

Having mentioned the principal geodesic operations which have, at different times, been effected in different countries, to determine the form of the earth, it remains only to indicate the manner in which the partial results deduced from those operations have been combined, in order to obtain the final result, which is, at present, regarded as most approximate to the existing reality. Three distinct modes have been successively adopted for arriving at the proposed end. As the result of inexact observations, and an incomplete theory, it was first assumed that the earth's figure was perfectly spherical. The labors of Picard, and of the French geometers, who immediately followed him, conclusively demonstrate the fallacy of this supposition; since, in contradiction of such an hypothesis, very different terrestrial radii were found to result from the several degrees of meridian measured, and within limits too wide to admit of the inference that these differences were, collectively, attributable to errors of observation, or mistakes in calculation.

The laws even then recognized, of the universal attraction of matter, the aspect of certain planets, such as Jupiter, which exhibit a flattening towards

*Sr. Vasquez Queipo: Disquisition on the Discourse of Señor Saavedra Meneses, before cited.

their poles, or the extremities of the axis of rotation, and an induction founded on well-proved facts, showing that the terraqueous globe existed, at some very remote period, in a state of perfect fluidity, furnished sufficient grounds for concluding that the earth, instead of being spherical, would naturally present an elliptical figure, or one slightly depressed in the direction of the polar axis. This being conceived, it remained simply to deduce from geodesical operations the value of the depression, or, what amounts to the same thing, the relation of the two axes of the generating ellipsoid, as well as the definite dimensions of those axes, for all which it had, in strictness, sufficed to measure two small arcs of meridian in widely separated latitudes-one, for instance, near the equator, the other in some inhabitable region nearest to either pole; nor, on the above supposition, would it have been of consequence whether those arcs corresponded to the same or to different meridians, while any intermediate arc, which might be measured, would serve for the verification of the former, as well as of the law of ellipticity, assumed as a point of departure. When the results of the scientific expeditions to Peru and Lapland were known, and were compared in the proposed view with those obtained in France, and for the first time the values of the oblateness of the earth and of the equatorial and polar axes were deduced, it was observed, not without surprise, that between the final deductions drawn, with the aid of so much experience, and with the theoretical ideas generated by those laborious investigations, there did not exist all the conformity which had been hoped for. The discordance, however, was at once attributed, not so much to the defect of the theory, as to the errors, to a certain extent inevitable, which had been committed in the course of the operations, or to local irregularities in the surface of the earth; but, as time advanced, and instruments were improved, while the obstacles already overcome served as useful indications to succeeding observers, the conviction was acquired, either that the form of the earth was not so simple and regular as was at first supposed, or, more probably, that the heterogeneity of its mass, and the inequality of the thickness of its crust, acting as disturbing causes, embarrassed the labors of geodesy, and opposed their indefinite advancement. Certain it is, at any rate, that at the close of the last century, as has been already intimated, great indecision prevailed as to the real value of the earth's ellipticity, and that, but for the resort to an ingenious mode of eluding the difficulty, the same doubt would have prevailed on this point to the present day. A single citation will prove the truth of what has been just said. The Russian general, Schubert, a distinguished mathematician and astronomer, collated, in a memoir published at St. Petersburg in 1859,* the elements of the eight principal arcs of meridian yet known, being the Russian arc, measured by Hansteen, Sclander, Struve, and Tenner; the Prussian, the English, and the French arcs; the arc measured in Pennsylvania by Mason and Dixon; that in Peru, by the Franco-Spanish commission; that in India, by Lambton and Everest; and that measured at the Cape of Good Hope by Maclear. By combining these eight ares, two. by two, in all possible manners, the Russian savant deduced, for the elements of the terrestrial ellipsoid, twenty-eight different results, between limits mach wider, doubtless, than the reader would imagine. Limiting ourselves,. for example, to the polar compression, the twenty-eight valuations just oited group themselves in this manner: Three are higher than the fraction; four are higher than, and lower than the preceding fraction; nine are comprised between the last and; seven between that and three between and 330; and two, finally, being those corresponding to the combinations of the Russian with the Prussian arc, and of the arc of the Cape with that of Pennsylvania, are lower than the fraction Supposing even that there were good and sufficient reasons for subtracting from the extreme values, it will still be

* Essai d'une Determination de la Veritable Figure de la Terre.

seen from this slight analysis of Schubert's work that there is a wide field for the exercise of doubt.

If it be conceded that the spherical figure of the earth is not admissible, and the elliptical appears as little accordant with the most probable results of observation, what other geometrical type will represent, better than these two, or more approximately than the second, the general form of our globe? None, in fact; for neither the more complicated figure, which Bouguer imagined, nor the idea of separating the axis of symmetry from the polar axis, suggested by Kligel, conceptions, both of them, which the theory of the attraction and primeval Hluidity of the earth excludes, are found to be exempt from the grave inconveniences which oppose themselves to the adoption of the second supposition. Of this truth Schubert himself supplies us with a good proof. In his memoir above cited, after analyzing the divergences, with reference to the form of the earth, according to the elements from which that form is deduced, and investigating the causes from which so great a discordance might proceed, he concludes by maintaining that the earth resembles, not so much an ellipsoid of revolution, as an ellipsoid of three axes, or, what is the same thing, that the meridians are be regarded as unequal ellipses, and the equator and parallels as also ellipses, and not as circles, as had, till that date, been believed. But the same astronomer, who seems so well persuaded of this consequence from his first investigations in April, 1859, affirms, in January, 1861,* that, setting aside the arc of India, he does not find, in the rest of the geodesic operations, any grounds for doubting that the terrestrial globe is an ellipsoid of revolution, compressed in the direction of the poles. What does this change of opinion, this vacillation, in a man of Schubert's merit prove, if not that this last figure represents that of the earth, as far as a geometrical abstraction can represent the forms. full of life and harmonious adaptability, of natural objects?

But, admitting the elliptical form, it still remains to determine its constitutive elements, and its dimensions; and, with this view, what is the combination of arcs of meridian which should be preferred to the rest, whether for the precision with which those arcs have been measured, the merit of the geometers to whom the operations were intrusted, or the favorable circumstances of time and territory in which they were executed? No single combination whatever: First. Because all that astronomers of merited reputation and conscientiousness profess to have done should be considered to be well done, or, at least, to be comparable with what other astronomers, endowed with the same qualities, are capable of realizing, under the penalty of introducing into the science a principle of endless confusion. Secondly. Because the differences which occur in the elements of the terrestrial ellipsoid, taken by separate combinations of arcs of meridian, indicate, not so much a defect in the operations, or a fault in the observers, as a real irregularity in the form of the earth, or the existence of disturbing causes, such as the local attraction of mountains, and even those, scarcely avoidable in practice, which proceed from the unequal density and thickness of a plane surface. And thirdly. Because if, in all strictness, the form which we seek does not coincide with the preconceived figure, the interests of truth will always vindicate their claim to recognition, if not by an apparent simplicity, at any rate by other more fertile qualities than pertain to any theory, however simple and seductive. In order, then, to deduce the geometrical figure of the earth the proper course would seem to be to take into view all the partial measurements which have been made, or such, at least, as are distinguished by some notable circumstance, as the place to which they correspond, the extent they embrace, or the accuracy which bas marked their execution, rejecting, of course, all which manifest carelessness on the part of the observers, or defect in the instruments which they have been obliged to employ; and, assuming that the ellipsoid of revolution is,

* Astronomische Nachrichten, No. 1303.

in theory, and to a certain point also by experiment, the hypothetical figure most conformable to reality, the final problem, one of pure mathematical analysis, and not certainly exempt from difficulties, will consist in finding, by a collation. of the values of the several arcs of meridian and parallel already measured, or hereafter to be measured, the curvature and dimensions of the ellipsoid of the above species, which, without exactly satisfying one or two geodesical operations, represents the results of all with the closest possible approximation. In this difficult labor the Germans, Walbeck and Schmidt, by combining, respectively, six and seven degrees of meridian, Bessel ten, Airy fourteen of meridian and four of parallel, and, finally, the Englishman, Colonel H. James, eight arcs of the former kind, which afforded the greatest assurance of exactness, arrived independently at results closely coinciding with one another, each of which might serve, in the absence of the rest, for a definite solution of the problem with which we are occupied. In the first of the two following tables, taken, though not entire, nor in the form here presented, from the Annual (Jahrbuch) of the Observatory of Berlin, for 1852, are shown the principal values given by the above mathematicians, together with the elements of the ellipsoid, which served for the establishment of the decimal metric system, in the calculation of which, as was before said, only the results obtained in Peru, France, and Lapland were taken into account, and that, too, before these were competently known. In the second table are presented other values, relative likewise to the form and volume of the earth, deduced from the fundamental elements of the globe, calculated by Bessel, and not less worthy of attention than those contained in the former table. The initials employed in both tables signify as follows:

In the first, R and r, the equatorial and polar radii; D, their difference; C, the polar compression of the globe, or the difference of the radii referred to the greater; e, the square of the eccentricity of any meridian ellipse, or, say, the difference of the squares of the two principal radii, referred to the square of the equatorial radius; Q and q, the values of the equatorial and meridian quadrants; and D and d, the values of a single degree of the equator and of a mean degree of meridian, computed in metres like all the preceding which do not express abstract relations.

In the second table the sign y marks the latitude or distance from the equator of the place or point to which the numbers on the right refer; M expresses the value of an arc of meridian of a single degree, comprised between the first and the corresponding latitudes of the margin; P, that of a degree of parallel; R, the terrestrial radius or distance of the surface from the centre of the earth variable with the latitude; and A, the area in square kilometres, comprised between two meridians separated by a degree of the equator and two parallels, between which intervenes a degree of meridian for different latitudes.

Elements of the terrestrial ellipsoid.

TABLE 1.

Walbeck, Schmidt, Bessel, Airy, James, (1799.) (1819.) (1829.) (1841.) (1849.) (1858.)