Figure of the ring, the motion of the the Earth before,' Huygens's investigation of the Earth's form. J. Cassini's Meridian measure. Measure of an arc in Peru. though less be retrograde; and the transitionan of Darien; and sailed to Peru. The difficulty of mea- History. this case to that of an oblate spheroid is sufficiently obvions The calculation which Newton has attempted this reasoning is in some respects erroneous; but the explanation is, perhaps, one of the strongest proofs genius. Indeed, if at this time we might presume to select the part of the Principia which probably astonished, and delighted, and satisfied its s readers more than any other, we should fix without hesitation on the explanation of the precession of the Equinoxes. 素 suring an are of parallel, and its inutility, (as on any In 1690 was published Huygens's Treatise. Dere Quito, and near the Southern extremity; and as base of Causa Gravitatis. It contains an investigation of the of the Earth, supposing the attraction upon every particle to be directed towards the centre, and to be always the same at equal distances from the centre. This supposition, it will be remarked, is directly opposed to one part of the Principle of Gravitation, namely, that which states' that every particle attracts every other particle. The ratio of the axes, is found in this manner to be 578 579.f 4110813617 Dos ng as to In 1684 J.D. Cassini (second of that name) commenced a trigonometrical measure of an an are of merkit dian in France, taking Picard's base as a foundation and proceeding Southward. It was intended that this should serve as a basis for a map of France and more accurate measure which was afterwards executed the same same line was used for that purpose, Some interruption took place, death of the Minister Colbert, but it length finished in 1701. The was at 9n't series of triangles' was then extended Northward to Dunkirk, kirk, and this part was finished in 1718. A base of 7246 toises was near Perpignan, and one of 5461 toises near Dunkirk. The difference of Tati toiseshe tudes of Paris and the Southern extremity, Collioure, en found to dbe 618 57", and her he' e distance 360,614 toises whence 157,097. The difference of latitudes of Paris and Dunkirk was found to be 2° 12′ 9′′.5, and the distance 125,468 toises; whence 1° 56,960. It appeared, then, that the degrees shortened in going was verification of 5259 toises near the Northern extremity. No other measure has been conducted under such extraordinary circumstances of locality The lowest point of their are was at an elevation of a mile and a half above the level of the sea; and, in some instances, the heights of two neighbouring signals differed more than a mile. At some places the danger was considered by the inhabitants to be so great, that public prayers were put up in the churches for their safety. The indolence and itl will of the natives troubled them much they were eveti in danger of losing their lives in a popular tumult; and to crown the whole, the instruments by which they were to determine their difference of latitude were found to be frot trustworthy, and the correétions of the places of the stars were not to be relied on. The former diffi culty they overcame by an ingenious reconstruction of their instruments, the latter by simultaneous observas tions at the two extremities of the arc. Their final result was that the whole length of the meridian arc, reduced to the level of the lowest station, was 176,945 toises, corresponding to a difference of latitudes of 3071"; and, therefore, that the length of an are of one degree was 56,767, or reduced to the level of the seas 56,748. The arc was extended about twenty minutes by Godin; but this addition is generally supposed to have been less accurately measured than the other parts. In this expedition an attempt was made by Bouguer to fouth to the North; and this seemed to indi- observe the effect of the attraction of Chimborazo of cate that the Earth was a prolate spheroid, the ratio of its axes being nearly 95: 96. A conclusion in such direct opposition: to Newton's theory, how extensively received, excited a great sensation among the Mathe maticians of Eumpe. The saccuracy of the measure and the certainty of the conclusion were sustained by Cassini; who, as well as his father and his son, seem to have been firmly convinced that the Figure of the Earth was elongated.To settle the point, a degree of a parallel was measured in 1734, extending from Strasburg to St. Malo. This, also, when compared with the others, seemed to lead to the sanie conclusion. It was alleged, however, that the difference of longitude of these stations was determined only by some ancient eclipses of Jupiter's satellites, and that no reliance could be placed on them. To escape from this state of doubt, it was determined that an are of meridian, and if neces sary, one-of parallel, should be measured near the Equator, in the comparison of which with the arc mea sured in France, the inevitable errors of observation would produce a smaller effect; and this is the origin of the celebrated expeditions of the French Academicians.. Every facility being afforded by Louis XV., in May, 1735, MM, Bouguer, Godin, Lacondamine, and others, sailed for South America; they then crossed the Isthmus Before the return of this party (who, in consequence® › of their dissensions, separated as quickly as possible, and regained Europe by different ways) another meat sure had been imagined and completed. Maupertuis, Clairaut, Camus, Lemonnier, and Outhier reached :the Gulf of Bothnia in July 1786. They had expected Measure to be able to make use of the islands in the Gulf for an are in the stations of a trigonometrical measure; but they, Sweden. found them so low, and so near the shore, that they were obliged to give this part of their the valley of the river ver Fornea, which passes by the town of the same name, seemned to offer a favourable line of country; and this was accordingly chosen. The stations (as in the Peruvian measure) were on the hills on each side; and the river gave them some facilities for But From this remark we ought, perhaps, to except a part of the survey of an are of parallel made in the present century, in which the principal chain of the Alps was passed, to nether, ne t " " tain Kittis, which was fixed on for the Northern extre mity. The latitudes were observed with a sector made by Graham. On their return to Tornea, the river being frozen, a base was measured on the ice nearly in the mid-a die of the arc, but no base of verification was measured. The result being somewhat different from what they had expected, the latitudes were reobserved, and some angles of the triangles, which had before been omitted, were observed. The final conclusion was that the dif ference of latitudes was 57 29,6, and the length of the arc: 55,923 toises; whence an are of 1° 57,422. In this expedition, as well as in that to Peru, observations were made to ascertain the length of the pendulum vibrating seconds; and they all agreed in showing that Gravity increased in going from the Equator towards the Pole. For a complete account of this measure wen must refer the reader to Maupertuis, La Figure de la Terreno 919 shutoul to 99th Todt nimalob of Bessure Ermessure-Before of an ellipticity of nearly tions 911 9-30 axes was E 7217 178179. This History. the Fure of the conveyance of their instruments. In the prosecuto ; that of the Peruvian and the Earth tion of their work great inconvenience was sustained m01 090814120one from the cold, the birds of prey, &el, but the greatest of do ngritur body stego us to in of ges all from the tormenting flies and gnats. The dogs also, Swedish arcs gave for its value. If, then, the meawith which some of the hills were continually en Earth's veloped increased the difficulties of the observations; sures were accurate, it was impossible that the in some instances they waited ten days on the top of a Figure could be elliptic: and Bouguer actually proposed, hill before they could see one of their signals. After as the only hypothesis which would satisfy observations, that the increas sixty-three days of great fatigue they reached the moun- of the lengths of degrees was proportignal to the the powers of the sines of latitude, and that the proportion of the would indicate a Figure projecting at middle latitudes above the elliptic spheroid with the same axes, To avoid interruption, in our account in our account of several imwe have omitted to mention some inportant measures, teresting deductions from the theory of Gravitation made about the same time. These homowe shall now describe. The first important to the theory of the * Maclaurin's Earth was made by Maclaurin. The Academy of Paris investiga proposed as a the ortay Essay in the yeaEarth's 170, The lembert, Maclaurin: the last by Euler, and is generally rally considered as one of the most elegant Geometrical investigations that was ever ever made. Most of the operaapplied, with very slight alterations, to the investigation of the Figure of the Earth, supposing it a homogeneous fluid; and the author has indicated the 1 Before the return of Bouguer and Lacondamine, the Principal results which can be obtained by this application. In this Treatise it was proved for the first time, whole of the French are of meridian was remeasured-by the oblate spheroid is a C's Ja Cassini, his son Cassini de Thury, and Lacaille. The a form of equilibrium; and he equation was given by which the ellipticity can, be operations of this measure are the subject of the Worki won entitled La Méridienne vérifiée. On measuring new the Equator to Gravity is known. This equation is found when the proportion of the centrifugal force at bases and making new observations of every kind, the cause of the original difficulty was soon discovered. transcendental; but by an approximate solution when the entrifugal force is small, it was shown that the The measure of Picard's base was erroneous by about ellipticity (that is the proportion of the excess of the Toth part of the whole, and this error had affected one & pars-only of the arc. Some inaccuracies both Astrono- equatorial diameter above the polar diameter to the 50BY mical and Geodetical were found to exist in Cassini's latter) is the ratio of the equatorial centrifugal measure. The late improvements in Astronomy, par,i) ale $14 od 1 bunot enw did bi to b ticularly the discovery of aberration and mutationed force to the equatorial gravity which in the case of enabled the new observers to give much greater exacted the Earth is 1 ness to the Astronomical observations They observed o the latitudes of five points, and thus obtained, in fact, fourlares of meridian Between Perpignan and Rodézto the length of an arc of one degree was found 57,04815 toises; between Rodéz and Bourges 57,040; between Bourges and Paris 57,071 and between Paris and Dunkirk 57,084. In the course of this measure an are of parallel was measured across the mouth of the In 1743 was published Clairant's Figure de la Terre. Clairaut's Rhone. From a station near Cette in Languedoc, and This is the most valuable Work that has been written investiga from Mont St. Victoire in Provence, the explosion of upon this subject. The same theory has since engaged tion. guapowden on a church between them was observed the attention of the most distinguished of the modern and from the difference of apparent times the difference Mathematicians, and finally of Laplace; and though More of of Bongitudes was found to be 1953 19 The length the form of the investigations has been much varied, of of the arc of: parallel in latitude 43°32 included: be- very little has been added to the results. The nature tween the two meridians was 76,599.6 toises. Theis of the equilibrium of fluids is first explained. The only comparison of this are with thebarc, of meridian from condition which had formerly been assumed as sufficient Perpignan to Rudéz gave, an ellipticity - surface should be perpendicular to the surface, or that of beguda grow the equilibrium of fluids was, that the force at the 1680 to vollez sd The comparison of any two of the three arcs in Peru, the pressure produced by the fluid in two canals of given in France, considering the whole as one arc, and in form, terminated at one extremity in the same point, and Sweden, showed that the degrees increased in goingon at the other extremity in the surface, should be the same. towards the Fole, and that, consequently, the Earth's Bouguer first showed that both of these conditions were Figure was flattened at the Poles. But when all three necessary, or something equivalent to both. Clairaut!! were considered, a singular, difficulty occurred, which has shown upon what the possibility of equilibrium we may be permitted to say is not yet wholly removed. depends; and he has applied these principles to the The comparison of the Peruvian and French arcs gave discovery of the form of equilibrium of a fluid on the the Earth. Clairaut's theorem, Nutation 5 2 Figure of lowing suppositions: 1st, that the Fluid is homogeneous, with a spheroidal nucleus of different density; 2dly, that the whole mass is fluid and heterogeneous. The form which makes equilibrium possible in all the variety o. cases which these suppositions include, is, approximately, an elliptic spheroid; the ellipticity is different according to the law of density, &c., but in all cases the following theorems are true: 1st, the increase of the length of degrees and of Gravity, in going from the Equator to the Poles, is as the square of the sine of latitude; 2dly, the sum of the ellipticity, and of the ratio of the whole increase of Gravity to the equatorial Gravity, is > the ratio of the centrifugal force at the Equator to the force of Gravity. The last of these theorems (one of the most important that has ever been discovered) is usually called by the name of its inventor. It is evident that, in conjunction with the former, it gives the means of determining the Earth's ellipticity from observations of the comparative force of Gravity at any two places. In 1747 Bradley discovered the nutation of the discovered Earth's axis. This had been alluded to by Newton as by observation and exa consequence of his theory, but no notice seems to plained by have been taken of his theoretical prediction. As soon, theory. however, as the fact was established, it was treated theoretically by Mathematicians: among the best of the Treatises upon nutation may be reckoned D'Alembert's Recherches sur la Précession, &c. Measure of In 1750 Boscovich and Le Maire, two Jesuits, Boscovich. measured an arc from Rome to Rimini. Their measures were made with the pace, the measure of the Country; but it was carefully compared with the French toise, so that their results could be expressed in terms of the same standard which had already served for so many measures. A base of 6139.5 toises was measured on the Via Appia; and a base of verification of 6037.6 toises by the sea-side near Rimini. The meridian distance was found 161,253.6 paces 123,221.3 toises.; and the difference of latitude 2° 9′ 47′′, whence 1° = 56,966.3 toises. This, when some corrections were applied, was reduced to 56,979; the mean latitude 42° 59' The whole of this measure passed over a mountainous country. Measure of Lacaille. Liesganig's measure. = In 1752 Lacaille, who had been sent to the Cape of Good Hope to make Astronomical observations, finding the circumstances of the country favourable for a trigonometrical survey, measured an arc of 1° 13′ 17.3, which he found 69,669.1 toises; whence 1° 57,037. The length of his base was 6467 toises. See Mém. de l'Acad. 1751. This arc presents a remarkable anomaly. According to this measure, a degree in the South hemisphere, whose mean latitude is 33° 20', is equal to a degree in the North hemisphere, whose mean latitude is about 45°. The known ability of the observer almost forbids the supposition of an error in the observations; and we have no grounds for conjecturing the cause of such a deviation from the law which seems to apply to the other arcs. The measures described in the next three paragraphs were undertaken, we believe, at the suggestion of Boscovich. In 1762 Liesganig, a Jesuit, began the measure of an arc of meridian passing through the Observatory of Vienna. See the Dimensio graduum Viennensis et Hungarici. Every care apparently was taken to ensure the correctness of this measure. The Vienna fathom was compared with a French toise furnished by Lacon damine and Lacaille; the iron quadrant of 23 feet, with History. which the angles of the triangles were observed, was repeatedly examined in all possible ways; the sector, for observation of latitude, (which we shall afterwards describe,) was on the most improved construction; the observations of latitude were sufficiently numerous; in every triangle, except one, Liesganig assures us, that the three angles were observed; and all the calculations were reexamined. The base (between Neustadt and Neunkirch) was of 6238 toises; the base of verification in Marchfeld was of 6388 toises. The concluded meridional distance of Sobieschiz and Varasdin (the North and South extremities) was 172,796 Vienna fathoms ; and the difference of latitudes 2° 56′ 45′′.5, whence 1° 58,655 Vienna fathoms = 57,077 toises. Yet in spite of all these apparent securities, there appears reason for rejecting this measure. The principal objections to it may be seen in Zach's Correspondance Astronomique, Reasons f &c. vol. vii. In the first place, it is certain that the star rejecting i which Liesganig has put down as μ Draconis is not μ Draconis, but some other star; possibly 85 Herculis, which Zach mentions. This would occasion no error, (as the same star was observe at different stations,) except in consequence of taking erroneous reductions. In the next place, Zach (who possessed some of Liesganig's manuscripts, and who recalculated some of the observations) affirms, that many of the observations had been altered to produce greater apparent agreement. And, lastly, it appears from a repetition of part of the survey by the Austrian officers in the beginning of the present century, that in one of Liesganig's triangles it was impossible, and apparently had always been so, to observe one of his stations from another; and, in fact, when the value of one of the angles, as given by him, was compared with the sum of two angles, which ought to be equal to it, observed by the.Austrian officers, a difference of 3° was found. This triangle is the last but two to the South; up to this the surveys agree within a few toises, but after this they sometimes differ in the situation which they assign to a station by more than 2500 toises. It would seem that a signal was mistaken; a similar accident happened to Snell; and there is reason to think that it has also occurred in the survey of England made under the direction of the Master-General of the Ordnance. Another are was measured by Liesganig on the plain of the Theiss, from Kistelech in the North to Czuroch, near Petervaradin, in the South. The length was 59,990 Vienna fathoms; and the difference of latitudes 1° 1' 34".5; whence 1° = 58,453 = 56,881 toises. Several circumstances prevent us from attaching much value to this measure; one is, the great number of small triangles which the nature of the country compelled him to use. About the same time Beccaria, a Jesuit, assisted by Beccaria another of the same Order, measured a degree in the measure plain of Lombardy. The measure of Boscovich crossed the Apennines, and its extremities were on the seacoasts; this of Beccaria (undertaken, as we have mentioned, at the suggestion of Boscovich) was carried over a flat country, and terminated at both extremities at the foot of lofty mountains. It was supposed that if the inequality of the country produced in the measure of a degree any sensible irregularity, the effects, in these two measures, would be of opposite kinds, and that the difference in the length of a degree would be such as to give a good idea of their magnitude. An iron toise, which had been compared by Lacondamine and Lacaille ure of with the toise used in the measure of the arc in Peru, Earth. (commonly called the toise of Peru,) was Beccaria's standard. His instruments were in almost every respect similar to those of Boscovich. The length of the base was 6501 toises; the meridional distance from Andratæ, the Northern extremity, to Mons Regalis, (Mondovi,) the Southern extremity, was 64,890 toises. The difference of latitude was found to be 1° 7′ 44′′.7; whence 1° 57,468 toises; a quantity much greater than other measures would have led us to expect. The mean latitude is 44° 57′. An account of this measure was pubits of lished under the title of Gradus Taurinensis. In the cy surveys made by the French officers during their occupation of this Country, and in the repetition of the observations for the latitudes of the stations by MM. Plana and Carlini, some discrepancies have been found, which would seem to show that the credit of this measure is rather doubtful. the attrac attraction of a mountain might be ascertained by direct History. sad In 1764 Messrs. Charles Mason and Jeremiah Dixon, 114 The anomalies in these measures were so considerable, that it appeared highly desirable to discover whether the attractions of mountains could have a sensible effect in altering the direction of Gravity. If in con sequence of the proximity of a mountain the plumb-line was drawn from the position which it would otherwise have maintained, it is evident that the point of the heavens which corresponded to the zenith of a station, as determined by Astronomical observations, would not be the same as if the mountain did not exist. If, for instance, a mountain were at L, near the point A, in fig. 1, and drew the plumb-line towards L, the apparent direction of Gravity, instead of A C would be A c, and the apparent horizontal line instead of AD would be Ad; the difference of latitudes, or the angle AC B, would be changed to A e B, and the length of the radius of curvature of the meridian would be changed from A C to Ac. On account of the influence of these effects on the determination of the Figure of the Earth, as well as for the important confirmation which might be given to the Principle of Gravitation, it was to be wished that the Schehallien and Schehallien was chosen. This is a mountain near the observa The arc of meridian which had been traced so accurately through France, served for a foundation to the surveys for a very accurate map of France. Nothing of this kind had been done in England, excepting a survey of part of the Highlands, commenced in consequence of the Rebellion of 1745. The disastrous wars of the last century prevented our Government from extending the map. In the year 1785, Cassini de Thury presented to the Junction of Royal Society a memorial on the uncertainty in the tions of difference of longitude of Greenwich and Paris; and pro- Greenwich posed that the English and French Mathematicians in and Paris. concert should determine, by Geodetic operations, the distance, measured along an arc of parallel. This was assented to, and the English survey was placed under the superintendence of General Roy, the French under that of Count Cassini, (fourth of that name,) Mechain, and Legendre. We believe it may fairly be said that in this, as in other grand experiments, though we began later than our continental neighbours, we conducted our operations with a degree of accuracy of which, till that time, no one had dared to form an idea. For the measure of the first base on Hounslow Heath, in 1784, deal rods Figure of (which had been used in all preceding measures) were the Earth. rejected, and glass tubes were used. These were afterwards laid aside, and a steel chain was employed in the measure of a base of verification on Romney Marsh. In 1791 the base on Hounslow Heath was remeasured with the chain. The description of the mode of measuring the bases (Phil. Trans. 1785, and Account of Trigonometrical Survey, vol. i.) was translated into French by M. Prony. The angles of the triangles were observed with a large theodolite, made by Ramsden; and this is the first instrument with which the spherical excess of the sum of the three angles above 180° (a quantity that rarely exceeds 3" or 4") was observed. The French part of this survey is remarkable also, as the first instance in which the newly-invented repeatingcircle was used in extensive operations. Whether this invention has contributed to the accuracy of Astronomical determinations, we may be permitted to doubt; but it can hardly be doubted that it is an excellent instrument for Geodetic operations. The chain of triangles formed by the English observers, extended from the neighbourhood of London to Dover, and across the channel to Calais, and some neighbouring stations; the French triangles joined these with the chain of triangles formerly observed in the meridian of Paris, in the neighbourhood of Dunkirk.* The object originally proposed in this survey was now attained; but it will readily be imagined that advantage was taken of such an admirable commencement for an equally excellent extension. An accurate survey of the whole Kingdom was commenced; several bases of verification were measured, and in the course of the survey, an arc of parallel between Beachy Head and Dunnose English are was measured in 1794, (by reciprocal observations of of parallel. azimuth, a method which we shall hereafter describe,) and an arc of meridian from Dunnose to Clifton in Yorkshire, in 1802. The latitudes were observed with English arc a large zenith sector made by Ramsden. The first of meridian. of these arcs gave for the length of a degree of longitude in the parallel of 50° 37' 7".3, 38,818 fathoms; the second gave for the length of a degree of meridian in latitude 52° 2', 60,820. This arc of meridian was divided into two parts, nearly equal, by the station of Arbury Hill; the arc from Dunnose to Arbury gave for one degree in latitude 51° 25', 60,864; that from Arbury to Clifton gave for one degree in latitude 52° 50', 60,766. These partial arcs present the same anomaly as those in France; the degrees appear to diminish in going towards the Pole. (See Account of Trigonometrical Survey, and Phil. Trans. for various years.) Origin of the new French measure. In 1791 the National Convention of France wishing to fix on a new standard of linear measure, determined (in the true spirit of the Revolutionary Philosophy) to select one which could not be considered as belonging to one nation rather than to another, but which might claim to be esteemed a standard for all the World. The length of the pendulum vibrating seconds at a given place had been pointed out by Picard as the best practical standard; but this did not suit the expanded ideas of the French legislators. They considered that the length of the seconds' pendulum varied on varying the place of experiment; but that, assuming the Earth's The whole of this measure has been repeated; the English part under the direction of Captain Kater, the French under that of M. Arago. See Phil. Trans. 1828. Forment surface (at least the Geometrical surface at which if History channels were made communicating with the sea the water would find its level) to be an exact surface of revolution, the length of the quadrant of meridian passing through every place would be the same. The Measure length of the quadrant of meridian passing through the arc from Observatory of Paris was therefore to be ascertained, Dunkirk and one ten-millionth part of this was to be called the mètre. It is almost unnecessary to add, that the idea of replacing a lost standard by means of an extensive Geodetic measure is perfectly chimerical; and that the only practical method (still subject to some uncertainty) is by ascertaining the length of the seconds' pendulum, which it is presumed, in the same place, does not vary from one Age to another. The measure was undertaken by Delambre and Mechain, and is described in the Base du Système Métrique; a Work which cannot be too strongly recommended to the perusal of all who wish to be acquainted with the methods pursued in an extensive survey by the ablest general observers and practical Mathematicians of the Age. The difficulties with which they had to struggle were very great, arising principally from the disturbed state of the Country, and, as they approached to a termination, from the depreciation of the assignats. Firesignals at night are preferable to all others for the operations of a survey; but these it was impossible to employ, as the people would have supposed them to be counter-revolutionary signals. In the choice of daysignals, and in procuring permission to erect aud to observe them, they experienced sometimes the most serious, and sometimes the most ludicrous obstacles. It was generally necessary to receive the sanction of the popular assembly of each town through which they passed; and in many cases Delambre found himself obliged to give to the collected inhabitants a sort of Lecture on Geodesy. When a great part of their labour was completed, they were dismissed from their office for not having sufficiently distinguished themselves by their hatred of Kings. They were, however, allowed to complete the arc which they had originally intended to measure; following the course of the former measures, and using, in many instances, the same signals from Dunkirk to Barcelona. The part South of the Pyrenees was measured by the permission and with the assistance of the Spanish Government. As soon as the details of the measures could be collected, they were laid before a committee consisting of scientific deputies from several continental nations. It was there determined to proceed on the assumption, that the Earth is an exact elliptical spheroid, and to calculate its ellipticity, its dimensions, and the length of the quadrant, from a comparison of the arc newly measured with the arc measured by Bouguer and Lacondamine in Peru. The ellipticity adopted was 1 and the mètre was fixed at 304' 443,296 lines. The latitudes were observed at Dunkirk, Paris, Evaux, Carcassone, and Montjouy, near Barcelona. The arc was thus divided into four partial arcs, of which the middle latitudes were 49° 56′, 47° 31', 44° 42', and 42° 17', and which gave respectively for the length of one degree, in toises, 57,082.7, 57,068.8, 56,977.8, and 56,946.6. This survey was afterwards extended by Biot and Arago to the Island of Formentera, near Minorca; the mean latitude of the additional arc was 40° 1', and the length of one degree 56,956.4. The mean latitude of the whole arc, from |