The Arabs, who were not inattentive to astronomy, did not overlook the question of the earth’s magnitude. The caliph Almamoum, 814 A.D., having fixed on a spot in the plains of Mesopotamia, despatched one company of astronomers northwards and another southwards, measuring the journey by rods, until each found the altitude of the pole to have changed one degree. But the result of this measurement does not appear to have been very satisfactory. From this time the subject seems to have attracted no attention until about 1500, when Fern el, a Frenchman, measured a distance in the direction of the meridian near Paris by counting the number of revolutions of the wheel of his carriage as he travelled. His astronomical observations were made with a triangle used as a quadrant, and his resulting length of a degree was by a happy chance very near the truth.

The next geodesist, Willebrυrd Snell, took an immense step in the right direction by substituting a chain of triangles for actual linear measurement. The account of this operation was published at Leyden in 1617. He measured his base line on the frozen surface of the meadows near Leyden, and measured the angles of his triangles, which lay between Alkmaar and Bergen-op-Zoom, with a quadrant and semicircles. He took the precaution of comparing his standard with that of the French, so that his result was expressed in toises (the length of the toise is about 6 - 39 English feet). The work was recomputed and reobserved by Muschenbroek in 1729.

In 1637 an Englishman, Richard Norwood, published his own determination of the figure of the earth in a volume entitled The Seaman's Practice, cοntayniηg a Fundamentali Probleme in Navigation experimentally verified, namely, touching the Compasse of the Earth and Sea and the quantity of a Degree in our English Measures. It appears that he observed on the 11th June 1633 the sun’s meridian altitude in London as 62° Γ, and on June 6, 1635, his meridian altitude in York as 59° 33 ’. He measured the distance between these places along the public road partly with a chain and partly by pacing. By this means, through compensation of errors, he arrived at 367,176 feet for the degree—a very fair result.

The application of the telescope to circular instruments was the next important step in the science of measurement. Picard was the first who in 1669, with the telescope, using such precautions as the nature of the operation requires, measured an arc of meridian. He measured with wooden rods a base line of 5663 toises, and a second or base of verification of 3902 toises; his triangulation extended from Mal voisine, near Paris, to Sourdon, near Amiens. The angles of the triangles were measured with a quadrant furnished with a telescope having cross-wires in its focus. The difference of latitude of the terminal stations was determined by observations made "with a sector on a star in Cassiopeia, giving 1° 22' 55'' for the amplitude. The terrestrial measurement gave 78,850 toises, whence he inferred for the length of the degree 57,060 toises.

Hitherto geodetic observations had been confined to the determination of the magnitude of the earth considered as a sphere, but a discovery made by Richer turned tho attention of mathematicians to its deviation from a spherical form. This astronomer, having been sent by the Academy of Sciences of Paris to the island of Cayenne, in South America, for the purpose of determining the amount of terrestrial refraction and other astronomical objects, observed that his clock, which had been regulated at Paris to beat seconds, lost about two minutes and a half daily at Cayenue, and that in order to bring it to measure mean solar time it was necessary to shorten the pendulum by more than a line. This fact, which appeared exceedingly curious, and was scarcely credited till it had been confirmed by the subsequent observations of Varin and Deshayes on the coasts of Africa and America, was first explained in the third book of Newton’s Principia, who showed that it could only be referred to a diminution of gravity arising either from a protuberance of the equatorial parts of the earth and consequent increase of the distance from the centre or from the counteracting effect of the centrifugal force. About the same time, 1673, appeared the work of Huyghens entitled De Horologio Oscillator™, in which for the first time were found correct notions on the subject of centrifugal force. It does not, however, appear that they were applied to the theoretical investigation of the figure of the earth before the publication of Newton's Principia. In 1690 Huyghens, following up the subject, published his treatise entitled De Causa Gravitatis, which contains an investigation of the figure of the earth on the supposition that the attraction of every particle is towards the centre.

Between 1684 and 1718 J. and D. Cassini, starting from Picard’s base, carried a triangulation northwards from Paris to Dunkirk and southwards from Paris to Collioure. They measured a base of 7246 toises near Perpignan, and a somewhat shorter base near Dunkirk; and from the northern portion of the arc, which had an amplitude of 2’ 12' 9", obtained for the length of a degree 56,960 toises; while from the southern portion, of which the amplitude was 6° 18' 57'', they obtained 57,097 toises. The immediate inference from this was that, the degree diminishing with increasing latitude, the earth must be a prolate spheroid. This conclusion was totally opposed to the theoretical investigations of Newton and Huyghens, and created a great sensation among the scientific men of the day. The question was far too important to be allowed to remain unsettled, and accordingly the Academy of Sciences of Paris determined to apply a decisive test by the measurement of arcs at a great distance from each other. For this purpose some of the most distinguished members of their body undertook the measurement of two meridian arcs—one in the neighbourhood of the equator, the other in a high latitude; and so arose the celebrated expeditions of the French Academicians. In May 1735, MM. Godin, Bouguer, and De la Condamine, under the auspices of Louis XV., proceeded to Peru, where, assisted by two Spanish officers, after ten years of laborious exertion they measured an arc of 3° 7' intersected by the equator. The second party consisted of Maupertuis, Clairaut, Camus, Lemonnier, and Outhier, who reached the Gulf of Bothnia in July 1736; they were in some respects more fortunate than the first party, inasmuch as they completed the measurement of an arc near the polar circle of 57'amplitude and returned to Europe within sixteen months from the date of their departure.

The measurement of Bouguer and De la Condamine was executed with great care, and on account of the locality, as well as the manner in which all the details were conducted, it has always been regarded as a most valuable determination. The southern limit was at a place called Tarqui, the northern at Cotchesqui. A base of 6272 toises was measured in the vicinity of Quito, near the northern extremity of the arc, and a second base of 5260 toises near the southern extremity. The mountainous nature of the country made the work very laborious, in some instances the difference of heights of two neighbouring stations exceeding a mile. The difficulties with which the observers had to contend were increased by the opposition of the more ignorant of the inhabitants, and they were at times in danger of losing their lives. They had also much trouble with their instruments, those with which they were to determine the latitudes proving untrustworthy. But their energy and ingenuity were equal to the occasion, and they succeeded by simultaneous observations of the same star at the two extremities of the arc in obtaining very fair results. The whole length of the arc amounted to 176,945 toises, while the difference of latitudes was 3° 7' 3”. In consequence of a misunderstanding that arose between De la Condamine and Bouguer, their operations were conducted separately, and each wrote a full and interesting account of the operation. Bouguer’s book was published in 1749; that of De la Condamine in 1751. The toise used in this measure was ever after regarded as the standard toise, and is always referred to as the Toise of Peru.

The party of Maupertuis, though their work was quickly despatched, had also to contend with great difficulties. They were disappointed in not being able to make use of the small islands in the Gulf of Bothnia for the trigonometrical stations, and were forced to penetrate into the forests of Lapland. They commenced operations at Tornea, a city situated on the mainland near the extremity of the gulf. From this, the southern extremity of their arc, they carried a chain of triangles northward to the mountain Kittis, which they selected as the northern terminus. In the prosecution of this work they suffered greatly from cold and the bites of flies and gnats. The latitudes were determined by observations with a sector (made by Graham) of the zenith distance of a and δ Draconis. The base line was measured on the frozen surface of the river Tornea about the middle of the arc; two parties measured it separately, and they differed by about 4 inches. The result of the whole was that the difference of latitudes of the terminal stations was 57' 29" ’ 6, and the length of the arc 55,023 toises. In this expedition, as well as in that to Peru, observations were made with a pendulum to determine the force of gravity; and these observations coincided with the geodetical results in proving that the earth was an oblate and not prolate spheroid.

In 1740 was published in the Paris Mémoires an account, by Cassini de Thury, of a remeasurement by himself and Lacaille of the meridian of Paris. With a view to determine more accurately the variation of the degree along the meridian, they divided the distance from Dunkirk to Collioure into four partial arcs of about two degrees each, by observing the latitude at five stations. The anomalous results previously obtained by J. and D. Cassini were not confirmed, but on the contrary the length of the degree derived from these partial arcs showed on the whole an increase with increasing latitude. In continuation of their labours, Cassini and Lacaille further measured an arc of parallel across the mouth of the Rhone. The difference of time of the extremities was determined by the observers at either end noting the instant of a signal given by flashing gunpowder at a point near the middle of the arc.

While at the Cape of Good Hope in 1752, engaged in various astronomical observations, Lacaille measured an arc of meridian of 1° 13' 17'', which gave him for the length of the degree 57,037 toises—an unexpected result, which has led to the modern remeasurement of the arc by Sir Thomas Maclear.

Passing over the measurements made between Rome and Rimini and on the plains of Piedmont by the Jesuits Boscovich and Beccaria, and also the arc measured with deal rods in North America by Messrs Mason and Dixon, we come to the commencement of the English triangulation. In 1783, in consequence of a representation from Cassini de Thury on the advantages that would accrue to science from the geodetic connection of Paris and Greenwich, General Roy was with the king’s approval appointed by the Royal Society to conduct the operations on the part of this country,—Count Cassini, Mechain, and Delambre being appointed on the French side. And now a precision previously unknown was brought into geodesy by the use of Ramsden’s splendid theodolite, which, was the first to make the spherical excess of triangles measurable. The wooden rods with which the first base was measured were speedily replaced by glass rods, which again were rejected for the steel chain of Ramsden. The details of this operation are fully given in the Account of the Trigonometrical Survey of England and Wales. Shortly after this, the National Convention of France, having agreed to remodel their system of weights and measures, chose, as applicable to all countries, for their unit of length the ten-millionth part of the meridian quadrant. In order to obtain this length precisely, the remeasurement of the French meridian was resolved on, and deputed to Delambre and Mechain. The details of this great operation will be found in the Base du Système Métrique Décimale. The arc was subsequently extended by MM. Biot and Arago to the island of Iviza.

The appearance in 1838 of Bessel’s classical work entitled Gradmessung in Ostpreussen marks an era in the science of geodesy. Here we find the method of least squares, a branch of the theory of probabilities, applied to the calculation of a network of triangles and the reduction of the observations generally. This work has been looked on as a model ever since, and probably it will not soon be superseded as such. The systematic manner in which all the observations were taken with the view of securing final results of extreme accuracy is admirable. The triangulation, which is a small one, extends about a degree and a half along the shores of the Baltic in a N.N.E. direction. The compound bars with which he measured his base line may be understood by the following brief description. On the surface of an iron bar two toises in length is laid a zinc bar, both being very perfectly planed and in free contact— the zinc bar being slightly shorter than the iron bar. They are united at one end only, and as the temperature varies the difference of length of the bars as seen at the other end varies; this difference of length is a thermometrical indication whereby a correction for temperature can be applied to the bars so as to reduce their length to that at the standard temperature. The bars in measuring were not allowed to come into contact, but the intervals left were measured by the interposition of a glass wedge. The results of all the comparisons of the four measuring rods with one another, and with the standards, are elaborately worked out by least squares. The angles were observed with theodolites of 12 and 15 inches diameter, and the latitudes determined by means of the transit instrument in the prime vertical—a method much used in Germany. The formulae employed in the reduction of the astronomical observations are very elegant. The reduction of the triangulation waβ carried out in the most thorough manner,—the sum of the squares of all the actual theodolite observations being made a minimum. As it is usual now to follow this method (sometimes only approximately) in all triangulations where great precision is required, we here give a brief description of the method. The equations of condition of a triangulation are those which exist between the supernumerary observed quantities and their calculated values, for, after there are just sufficient observations to fix all the points, then any angle that may be subsequently observed can be compared with its calculated value. If a triangulation consist of n + 2 points, two of which are the ends of a base line, then to fix the n points 2n angles suffice; so that if m be the actual number of angles really observed, the triangulation must afford m - 2n equations of condition. To show how these arise, suppose that from a number m of fixed points A, B, C . . . a new point P is observed, which m points are again observed from P, then there will be formed m - 1 triangles, in each of which the sum of the observed angles is = 180° + the spherical excess; this gives at once m - 1 equations of condition. The m — 2 distances will each afford an equation of the form

PC PB PA

PB ’ PA PC ’

not, however, limited to three factors. Should P observe the m points and not be observed back, there will be m - 3 equations of the above form (they are called side equations). In a similar manner other cases can be treated. In practice the ratios of sides are replaced by the ratios of the sines of the corresponding opposite angles. To each observed angle a symbolical correction is applied, so that if α be an observed angle and a + x the true or most probable angle, sin (α + z) = sin α(1 + x cot a), x being a small angle whose square is neglected. Thus the side equation takes the form β + β λ x χ + β i x 2 + . . . β 1 x t = 0. In the case of equations formed by adding together the three observed angles of a triangle the co-efficients are of course unity. The problem then is this: Given n equations

β + 3 r r 1 + β. 2 x 2 + . . . β m r m = 0 β' + j 3 1 'z 1 + β 3 'x 2 + . . . βfx m = 0 β"+ βfx 1 + β i "x a + . . . β m "r m = 0

between m(m>n) unknown quantities x 1 . . .x m , which are the corrections (expressed in seconds of arc) to the observed angles, it is required to determine these quantities so as to render the function w 1 x 1 2 + tv 2 x 2 2 + w. 0 x 3 2 + . . . w m x m 2 a minimum, where w 1 . . . w m are the weights of the determinations of the angles to which the corresponding corrections belong. The corrections x l . . . x η fulfilling this condition of minimum have, according to the theory of least squares, a higher probability than any other system of corrections that merely satisfy the equations of condition. Multiply the n equations by multipliers λ 1 , λ 2 , . . . λ n , and we obtain by the theory of maxima and minima m equations w 1 x 1 ≈ β l λ 1 + β 1 ’ λ i + βfλ i + . . .

= 3aM ^t h 2 + β∙> λj + . l j wmrm — 3∙tλj + βmλ 2 + ß M λ-3 + . .

The values of x i . . . x m obtained from these equations are to be substituted in the original equations of condition, and then there will be n equations between the n multipliers λ 1 . . . λ n . These being solved, the numerical values of λ 1 . . . λ n will be obtained, and on substituting these in the last equations...