do this we must be guided by the weight of the determinations of each angle. When a series of direct and independent observations is made, under similar circumstances, of any measurable magnitude -as an angle-the weight of the result is equal to half the square of the number of observations divided by the sum of the squares of the differences of the individual measures from the mean of all. Now let h, k,l be the weights of the three measured angles, and let x, y, z be the corrections which should be applied to them. We know that x+y+z+=0; and the theory of probabilities teaches us that the most probable values are those which make hx2+ky2+lz2 a minimum. Here we arrive at a simple definite problem, the result of which is hx-ky-lz, showing that e has to be divided into three parts which shall be proportional to the reciprocals of the weights of the corresponding angles. In what follows we shall, for simplicity, suppose the weights of the observed angles to be all equal. Suppose now that A, B, C are the three angles of a triangle, and that the observed values are A+e, B+e, Č+e; then, although 1, 2, 3, the errors of observation, are unknown, yet by adding up the observed angles and finding that the sum is in excess of the truth by a small quantity e, we get +e+3=c. Now, according to the last proposition, if we suppose the angles to be equally well observed, we have to subtract je from each of the observed values, which thus become A+ že1 - je, - §ez, B-e1+c2-e3, C− že1 − }c2+}c3. Then to obtain a and b by calculation from the known side c, we have A spheroidal triangle differs from a spherical triangle, not only in that the curvatures of the sides are different one from another, but more especially in this that, while in the spherical triangle the normals to the surface at the angular points meet at the centre of the sphere, in the spheroidal triangle the normals at the angles A, B, C meet the axis of revolution of the spheroid in three different points, which we may designate a, ß, y respectively. Now the angle A of the triangle as measured by a theodolite is the inclination of the planes BAa and CAa, and the angle at B is that a sin (C — §o1⁄2 — §2+})=c sin (a + že, − §c, − §e3), contained by the planes ABB and CBB. But the planes with a similar expression for the relation between b and c. Put ABa and ABB containing the line AB in common cut the a, B, y for the cotangents of A, B, C, then the errors of the computed surface in two distinct plane curves. γ In order, therefore, values of a and b are expressed thusthat a spheroidal triangle may be exactly defined, it is necessary that the nature of the lines joining the three vertices be stated. In a mathematical point of view the most natural definition is that the sides be geodesic or shortest lines. Gauss, in his most elegant treatise entitled Disquisitiones generales circa superficies curvas, has entered fully into the subject of geodesic triangles, and has investigated expressions for the angles of a geodesic triangle whose sides are given, not certainly finite expressions, but approximations inclusive of small quantities of the fourth order, the side of the triangle or its ratio to the radius of the nearly spherical surface being a small quantity of the first order. The terms of the fourth order, as given by Gauss for any surface in general, are very complicated even when the surface is a spheroid. If we retain small quantities of the second order only, and put A, B, C for the angles of the geodesic triangle, while A, B, C are those of a plane triangle having sides equal respectively to those of the geodesic triangle, then, σ being the area of the triangle and a, b, c the measures of curvature at the angular points, 8b=§b{e2( − B + y) +e»( 2B+y) +eg( − B − 2y)} • Now these actual errors must remain unknown; but we here make use of the following theorem, proved in the doctrine of probabilities. The probable error of a quantity which is a function of several independently observed elements is equal to the square root of the sum of the squares of the probable errors that would arise from each of the observed elements taken singly. Now suppose that each angle in a triangle has a probable error e, then we replace 1, 2, 3 by e, and adding up the squares of the coefficients find for the probable error of a, ae √6 √(a2+ay+y2), and for that of b, ± be √6 √(ẞ2+By+y2). Suppose the triangle equilateral, each side eiglit miles, and the probable error of an observed angle 0"-3; then the probable error of either of the computed sides will be found to be 0-60 inches. + H K Take a chain of triangles as indicated in the diagram (fig. 5); suppose all the angles measured, and that the sides MN, HJ are measured bases; it is required to investigate the necessary corrections to the observed angles in order not only that the sum of the three angles of each triangle fulfil the necessary condition, but that the length r, of HJ, calculated from that of MN, M shall agree with the measured length. Y3 X3 Fig. 5. x 7.3 x 2 N Let X, Y, Z, &c., be the angles as observed, 1, 1, 1, &c., the required corrections; then each triangle on adding up the angles gives an equation 1+1+1+€1 =0. Let the corrected angles be X1=X+x, Y1-Y+y, &c., then HJ sin X sin X sin X sin X¦ == The geodesic line being the shortest that can be drawn on any surface between two given points, we may be conducted to its most important characteristics by the following considerations: let p, q be adjacent points on a curved surface; through s the middle point of the chord pq imagine in the intersection of this plane with the surface; then a plane drawn perpendicular to pq, and let S be any point PS+ Sq is evidently least when sS is a minimum, which is when sS is a normal to the surface; hence it follows that of all plane curves on the surface joining p, q, when those points are indefinitely near to one another, that is the shortest which is made by the normal plane. That is to say, the osculating plane at any point of a geodesic line contains the normal to the surface at that point. Imagine now three points in space, A, B, C, such that AB=BC=c; let the direction cosines of AB be l, m, n, those of BC l',m', n', then x,y,z being the coordinates of B, those of A and C will be respectively— x-cly-cm- en x+cl': y+cm' : z+cn'. Hence the coordinates of the middle poiut M of AC are x+c(l' −1), y + }c(m' — m), z + {c(nn), and the direction dx dy dz which, however, are equivalent to only one equation. d2x d2y =0, In Y dsa From this it may be shown that the azimuth at A of the The circumstance that the angles of the geodesic triangle Let the azimuths (at the middle point, say) of the sides BC, CA, A'-A-12 B'- B= cos 2(b2 sin 28 - c2 sin 2y), 127-2 cos 24(c sin 2y - a2 sin 2a), C-C 12,-2 cos 2p(a2 sin 2a – b2 sin 28). C1 = 120,2 12073). It is but seldom that the terms of the fourth order are required. Omitting them, we have Legendre's theorem, viz., "If from each of the angles of a spherical triangle, the sides of which are small in comparison with the radius, one-third of the spherical excess be deducted, the sines of the angles. thus diminished will be proportional to the length of the opposite sides, so that the triangle may be computed as a plane triangle." By this means the spherical triangles which present themselves in geodesy are computed with very nearly the same ease as plane triangles. And from the expressions given above for the spheroidal angles A', B', C' it may be proved that no error of any consequence can arise from treating a spheroidal triangle as a spherical, the radius of the sphere being as stated above. When the angles of a triangulation have been adjusted by the method of least squares, the next process is to calculate the latitudes and longitudes of all the stations starting from one given point. The calculated latitudes, longitudes,. and azimuths, which are designated geodetic latitudes, longitudes, and azimuths, are not to be confounded with the observed latitudes, longitudes, and azimuths, for these last are subject to somewhat large errors. Supposing the latitudes of a number of stations in the triangulation to be observed, practically the mean of these determines the posi tion in latitude of the network, taken as a whole. So the orientation or general azimuth of the whole is inferred from all the azimuth observations. The triangulation is then supposed to be projected on a spheroid of given elements, representing as nearly as one knows the real figure of the earth. Then, taking the latitude of one point and the direction of the meridian there as given-obtained, namely, from the astronomical observations there-one can compute the latitudes of all the other points with any degree of precision that may be considered desirable. It is necessary to employ for this purpose formula which will give results true even for the longest distances to the second place of decimals of seconds, otherwise there will arise an accumulation of errors from imperfect calculation which should always be avoided. For very long distances, eight places of decimals should be employed in logarithmic calculations; if seven places only are available very great care will be required to keep the last place true. Now let , ' be the latitudes of ø, two stations A and B; a, a' their mutual azimuths counted from north by east continuously from 0° to 360°; w their difference of longitude measured from west to east; and & the distance AB. ηπ sin 2a tan 1, cos (a-e) - n, Po s sin (a-e) =3 n cus (p'+n)' Here n is the normal or radius of curvature perpendicular | shown that this inclination, amounting ordinarily to one or to the meridian; both n and p correspond to latitude 1, and Po to latitude (+). For calculations of latitude and longitude, tables of the logarithmic values of p sin 1′′, n sin 1", and 2np sin 1" are necessary. The following table contains these logarithms for every ten minutes of latitude from 52° to 53° computed with the elements a= 20926060 and a:c=295: 294 : The logarithm in the last column is that required also for the calculation of spherical excesses, the spherical excess of a triangle ab sin C being expressed by 2pn sin 1′′ * AP: =scos (a-3), two seconds, may in some cases exceed 10, or, as at the foot of the Himalayas, even 30 seconds. By the expression "mathematical figure of the earth we mean the surface of the sea produced in imagination so as to percolate the continents. We see then that the effect of the uneven distribution of matter in the crust of the earth is to produce small elevations and depressions on the mathematical surface which would be otherwise spheroidal. No geodesist can proceed far in his work without encountering the irregularities of the mathematical surface, and it is necessary that he know how they affect his astronomical observations. The whole of this subject is dealt with in his usual elegant manner by Bessel in the Astronomische Nachrichten, Nos. 329, 330, 331, in a paper entitled "Ueber den Einfluss der Unregelmässigkeiten der Figur der Erde auf geodätische Arbeiten, &c." But without entering into further details it is not difficult to see how local attraction at any station affects the determinations of latitude, longitude, and azimuth there. η Let there be at the station an attraction to the north-east throwing the zeuith to the south-west, so that it takes in the celestial sphere a position Z', its undisturbed position It is frequently necessary to obtain the coordinates of being Z. Let the rectangular components of the displaceone point with reference to another point; that is, let a ment ZZ' be measured southwards and measured westperpendicular arc be drawn from B to the meridian of Awards. Now the great circle joining Z' with the pole of meeting it in P, then, a being the azimuth of B at A, the the heavens P makes there an angle with the meridian coordinates of B with reference to A are PZ = cosec PZ'n sec o, where is the latitude of the station. Also this great circle meets the horizon in a point whose distance from the great circle PZ is n sec o sin = n tan . That is, a meridian mark, fixed by observations of the pole star, will be placed that amount to the east of north. Hence the observed latitude requires the correction έ; the observed longitude a correction 7 sec ; and any observed azimuth a correction ʼn tan p. Here it is supposed that azimuths are measured from north by east, and longitudes eastwards. BP =s sin (a-e), where e is the spherical excess of APB, viz., s2 sin a cos a multiplied by the quantity whose logarithm is in the fourth column of the above table. Irregularities of the Earth's Surface. In considering the effect of unequal distribution of matter in the earth's crust on the form of the surface, we may simplify the matter by disregarding the considerations of rotation and excentricity. In the first place, supposing the earth a sphere covered with a film of water, let the density p be a function of the distance from the centre so that surfaces of equal density are concentric spheres. Let now a disturbance of the arrangement of matter take place, so that the density is no longer to be expressed by p, a function of only, but is expressed by p+p', where p' is a function of three coordinates 0, 4, r. Then p' is the density of what may be designated disturbing matter; it is positive in some places and negative in others, and the whole quantity of matter whose density is p' is zero. The previously spherical surface of the sea of radius a now takes a new form. Let P be a point on the disturbed surface, P' the corresponding point vertically below it on the undisturbed surface, PP'u. The knowledge of u over the whole surface gives us the form of the disturbed or actual surface of the sea; it is an equipotential surface, and if V be the potential at P of the disturbing matter p', M the mass of the earth, M M M As far as we know, u is always a very small quantity, and proximate estimate of the effect of a compact mountain in The expression given for u enables one to form an apraising the sea-level. Take, for instance, Ben Nevis, which contains about a couple of cubic miles; a simple calculation shows that the elevation produced would only amount to about 3 inches. In the case of a mountain mass like the Himalayas, stretching over some 1500 miles of country with a breadth of 300 and an average height of 3 miles, although it is difficult or impossible to find an expression for V, yet hundred feet may exist near their base. The geodetical we may ascertain that an elevation amounting to several operations, however, rather negative this idea, for it is shown in a paper in the Philosophical Magazine for August 1878 Indian arc departs but slightly from that of the mean figure by Colonel Clarke that the form of the sea-level along the of the earth. If this be so, the action of the Himalayas must be counteracted by subterranean tenuity. work of triangulation projected on or lying on a spheroid Suppose now that A, B, C, . . . are the stations of a netof semiaxis major and excentricity a, e, this spheroid having its axis parallel to the axis of rotation of the earth, and its surface coinciding with the mathematical surface of the earth at A. Then basing the calculations on the observed elements at A, the calculated latitudes, longitudes, and directions of the meridian at the other points will be the true latitudes, &c., of the points as projected on the spheroid. On comparing these geodetic elements with the corresponding astronomical determinations, there will appear a system of differences which represent the inclinations, at the various points, of the actual irregular surface to the surface of the spheroid of reference. These differences will suggest two things,-first, that we may improve the agreement of the two surfaces, by not restricting the spheroid of refer ence by the condition of making its surface coincide with the mathematical surface of the earth at A; and secondly, by altering the form and dimensions of the spheroid. With respect to the first circumstance, we may allow the spheroid two degrees of freedom, that is, the normals of the surfaces at A may be allowed to separate a small quantity, compounded of a meridional difference and a difference perpendicular to the same. Let the spheroid be so placed that its normal at A lies to the north of the normal to the earth's surface by the small quantity έ and to the east by the quantity 7. Then in starting the calculation of geodetic latitudes, longitudes, and azimuths from A, we must take, not the observed elements p, a, but for 4, 4+§, and for a, a+ŋ tan 6, and zero longitude must be replaced by n sec p. At the same time suppose the elements of the spheroid to be altered from a, e to a+da, e+de. Confining our attention at first to the two points A, B, let (p'), (a'), (w) be the numerical elements at B as obtained in the first calculation, viz., before the shifting and alteration of the spheroid; they will now take the form (p')+ƒ§+ gn+ hda +kde, (a) +ƒ'§ + g'n+h'da + k'de, (w) +f"&+g'n+h"da+k"de, where the coefficients f, g, ... &c. can be numerically calculated. Now these elements, corresponding to the projection of B on the spheroid of reference, must be equal severally to the astronomically determined elements at B, corrected for the inclination of the surfaces there. If έ', n' be the components of the inclination at that point, then we have § = ($') − $' + ƒ+ gn+ hda+kde, n' tan p' (a) a' + f '§ + g'n + h'da + k'de, n' sec p' (w). w+f"&+g"n+h"da+k"de, where ', a', w are the observed elements at B. Here it appears that the observation of longitude gives no additional information, but is available as a check upon the azimuthal observations. = = g earth are altered, and the effect upon the observation of latitude is a very small error expressed by the formula h g-g sin 24, where g, g' are the values of gravity at a the equator and at the pole. This is also a quantity which may be neglected, since for ordinary mountain heights it amounts to only a few hundreths of a second. The uncertainties of terrestrial refraction render it impossible to determine accurately by vertical angles the heights of distant points. Generally speaking, refraction is greatest at about daybreak; from that time it diminishes, being at a minimum for a couple of hours before and after mid-day; later in the afternoon it again increases. This at least is the general march of the phenomenon, but it is by no means regular. The vertical angles measured at the station on Hart Fell showed on one occasion in the month of September a refraction of double the average amount, lasting from 1 P.M. to 5 P.M. The mean value of the coefficient of refraction k determined from a very large number of observations of terrestrial zenith distances in Great Britain is 07920047; and if we separate those rays which for a considerable portion of their length cross the sea from those which do not, the former give k = '0813 and the latter k=0753. These values are determined from high stations and long distances; when the distance is short, and the rays graze the ground, the amount of refraction is extremely uncertain and variable. A case is noted in the Indian Survey where the zenith distance of a station 10.5 miles off varied from a depression of 4' 52"-6 at 4.30 P.M. to an elevation of 2' 24"-0 at 10.50 P.M. If h, h' be the heights above the level of the sea of two stations, 90° +8, 90°+8′ their mutual zenith distances (8 being that observed at h), s their distance apart, the earth being regarded as a sphere of radius = a, then, with sufficient precision, h'-h-s tan h-h's tan ($1-26-8), -8'). ( S 1-2k 2a If now there be a number of astronomical stations in the triangulation, and we form equations such as the above for each point, then we can from them determine those values of έ, n, da, de, which make the quantity 2+2+If from a station whose height is h the horizon of the sea ¿12+n22 + . spheroid which best represents the surface covered by the be observed to have a zenith distance 90° +8, then the above formula gives for h the value triangulation. a minimum. Thus we obtain that The precise determination of the altitude of his station is a matter of secondary importance to the geodesist; nevertheless it is usual to observe the zenith distances of all trigonometrical points. The height of a station does indeed influence the observation of terrestrial angles, for a vertical line at B does not lie generally in the vertical plane of A, but the error (which is very easily investigated) involved in the neglect of this consideration is much smaller than the errors of observation. Again, in rising to the height h above the surface, the centrifugal force is increased and the magnitude and direction of the attraction of the tan2 8 21-2k = Suppose the depression & to be n minutes, then h= 1.054n" if the ray be for the greater part of its length crossing the sea; if otherwise, h=1040n2. To take an example: the mean of eight observations of the zenith distance of the sea horizon at the top of Ben Nevis is 91° 4′ 48′′, or d= 64.8; the ray is pretty equally disposed over land and water, and hence h=1·047n2 = 4396 feet. The actual height of the hill by spirit-levelling is 4406 feet, so that the error of the height thus obtained is only 10 feet. Longitude. The determination of the difference of longitude between two stations A and B resolves itself into the determination of the local time at each of the stations, and the comparison by signals of the clocks at A and B. Whenever telegraphic lines are available these comparisons are made by electro-telegraphy. A small and delicately-made apparatus introduced into the mechanism of an astronomical clock or chronometer breaks or closes by the action of the clock a galvanic circuit every second. In order to record the minutes as well as seconds, one second in each minute, namely that numbered 0 or 60, is omitted. The seconds are recorded on a chronograph, which consists of a cylinder revolving uniformly at the rate of one revolution per minute covered with white paper, on which a pen having a slow movement in the direction of the axis of the cylinder describes a continuous spiral. This pen is deflected through the agency of an electromagnet every second, and thus the seconds of the clock are recorded on the chronograph by offsets from the spiral curve. An observer having his hand on a contact key in the same circuit can record in the same manner his observed times of transits of stars. The method of determination of difference of longitude is, therefore, virtually as follows. After the necessary observations for instrumental corrections, which are recorded only at the station of observation, the clock at A is put in connexion with the circuit so as to write on both chronographs, namely, that at A and that at B. Then the clock at B is made to write on both chronographis. It is clear that by this double operation one can eliminate the effect of the small interval of time consumed in the transmission of signals, for the difference of longitude obtained from the one chronograph will be in excess by as much as that obtained from the other will be in defect. The determination of the personal errors of the observers in this delicate operation is a matter of the greatest importance, as therein lies probably the chief source of residual error. Since the article FIGURE OF THE EARTH was written, considerable additions to the data for the determination of the semiaxes of the earth have been obtained from India, viz., a new meridian arc of 20°, the southern point of which is at Mangalore, together with several arcs of longitude, the longest of which, between Bombay and Nizagapatam, extends over 10° 30'. The effect of the accession of these new measures is to alter the figure previously given to the following: the semiaxes of the spheroid best representing the large arcs now available are α= 20926202; C= =20854895; c: a=292 465: 293-465. This value of the major semiaxis exceeds that previously given by 140 feet, whereas the new polar semiaxis is less than the old by 226 feet. If we admit that the figure may possibly be an ellipsoid (not of revolution), then the investigation leads us, through the solution of 51 equations, to these values of the semiaxes a=20926629, b=20925105, C= - 20854407. The greater axis of the equator lies in longitude 8° 15' west of cuts off a portion of the north-west corner of Africa, and in the Greenwich, a meridian which passing through Ireland and Portugal opposite hemisphere cuts off the north-east corner of Asia. The apparent ellipticity of the equator is much reduced by the addition of the new data, and it would not be right to put too much confidence in the ellipsoidal figure until many more arcs of longitude shall have furnished the nieans of testing the theory more decisively than can be done at present. (See Philosophical Magazine, August 1878.) (A. R. C.) GEOFFREY OF MONMOUTH (1110-1154), one of the most famous of the Latin chroniclers, was born at Monmouth early in the 12th century. Very little is known of his life. He became archdeacon of the church in Monmouth, and in 1152 was elected bishop of St Asaph. He died in 1154. Three works have been attributed to him—the Chronicon sive Historia Britonum; a metrical Life and Prophecies of Merlin; and the Compendium Gaufredi de Corpore Christi et Sacramento Eucharistic. Of these the first only is genuine; internal evidence is fatal to the claims of the second; and the Compendium is known to be written by Geoffrey of Auxerre. The Historia Britonum appeared in 1147, and created a great sensation. Geoffrey professed that the work was a translation of a Breton work he had got from his friend Walter Calenius, archdeacon of Oxford. It is highly probable that the Breton work never existed. The plea of translation was a literary fiction extremely common among writers in the Middle Ages, and was adopted to give a mysterious importance to the communications of the author and to deepen the interest of his readers. We may compare with this Sir Walter Scott's professed quotations from "Old Plays," which he wrote as headings for chapters in his novels. If Geoffrey consulted a Breton book at all, it would probably be one of the Arthurian romances then popular in Armorica. His history is a work of genius and imagination, in which the story is told with a Defoe-like minuteness of detail very likely to impose on a credulous age. It is founded largely on the previous histories of Gildas and the so-called Nennius; and many of the legends are taken direct from Virgil. The history of Merlin, as embodied in the Historia, is found in Persian and Indian books. Geoffrey's imagination may have been greatly stimulated by local English legends, especially in the numerous stories he gives in support of his fanciful derivations of names of places. Whatever hints Geoffrey may have got from popular tales, and whatever materials he may have accumulated in the course of his reading, the Historia is to be thought of as largely his own creation and as forming a splendid poetical whole. Geoffrey, at all events, gave these stories their permanent place in litera ture. We have sufficient evidence to prove that in Wales the work was considered purely fabulous. (See Giraldus Cambrensis, Itinerarium Cambria, lib. i., c. 5, and Cambrie Descriptio, c. vii.) And William of Newbury says "that fabler (Geoffrey) with his fables shall be straightway spat out by us all." Geoffrey's Historia was the basis of a host of other works. It was abridged by Alfred of Beverley (1150), and translated into Anglo-Norman verse, first by Geoffrey Gaimar (1154), and then by Wace (1180), whose work, Li Romans de Brut, contained a good deal of new matter. Early in the 13th century was published Layamon's Brut; and in 1278 appeared Robert of Gloucester's rhymed Chronicle of England. These two works, being written in English, would make the legends popular with the common people. The same influence continued to show itself in the works of Roger of Wendover (1237), Matthew Paris (1259), Bartholomew Cotton (1300), Matthew of Westminster (1310), Peter Langtoft, Robert de Brunne, Ralph Higden, John Harding, Robert Fabyan (1512), Richard Grafton (1569), and Raphael Holinshed (1580), who is especially important as the immediate source of some of Shakespeare's dramas. A large part of the introduction of Milton's History of England consists of Geoffrey's legends, which are not accepted by him as historical. The stories, thus preserved and handed down, have had an enormous influence on literature generally, but especially on English literature. They became familiar to the Continental nations; and they even appeared in Greek, and were known to the Arabs. With the exception of the translation of the Bible, probably no book has furnished so large an amount of literary material to English writers. The germ of the popular nursery tale, Jack the Giant-Killer, is to be found in the adventures of his Corineus, the companion of Brutus, who settled in Cornwall, and had a desperate fight with giants there. Goëmagot, one of these giants, is said to be the origin of Gog and Magog-two effigies formerly exhibited on the Lord Mayor's day in London, which are referred to in several of the English dramatists, and still have their well-known representatives in the Guildhall of the city. Chaucer gives Geoffrey a place in his "House of Fame," where he mentions "Englyssh Gaunfride" (Geoffrey) as being "besye for to bere up Troye." Meanwhile the Arthurian romances had assumed a unique place in literature. The Arthur of later poetry is a grand ideal personage, seemingly unconnected with either space or time, and performing feats of extraordinary and superhuman valour. The real Arthur-if his historical existence is to be conceded--was most probably a Cumbrian or |