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each in mean latitudes, and near each other, say about five |
degrees of latitude apart, the probable error of the resulting
value of the ellipticity will be approximately 800€, €
being expressed in seconds, so that if be so great as 2"
the probable error of the resulting ellipticity will be greater
than the ellipticity itself. It is not only interesting, but
necessary at times, to calculate the attraction of a mountain,
and the consequent disturbance of the astronomical zenith,
at any point within its influence. The deflection of the
plumb-line, caused by a local attraction whose amount is
AS, is measured by the ratio of A8 to the force of gravity
at the station. Expressed in seconds, the deflection A is
A=12".417. A,

P

cedure. Draw on the contoured map a series of equidistant circles, concentric with the station, intersected by radial lines so disposed that the sines of their azimuths are in arithmetical progression. Then, having estimated from the map the mean heights of the various compartments, the calculation is obvious.

In mountainous countries, as near the Alps and in the Caucasus, deflections have been observed to the amount of as much as 29". On the other hand, deflections have been observed in flat countries, such as that noted by Professor Schweitzer, who has shown that, at certain stations in the vicinity of Moscow, within a distance of 16 miles the plunbline varies 16" in such a manner as to indicate a vast deficiency of matter in the uuderlying strata. But these are exceptional cases. Since the attraction of a mountain mass is expressed as a numerical multiple of 8: p, the ratio of the density of the mountain to that of the earth, if we have any independent means of ascertaining the amount of the deflection, we have at once the ratio p: 8, and thus we obtain the mean density of the earth, as, for instance, at Schiehallion, and more recently at Arthur's Seat. A conpact mass of great density at a small distance under the surface of the earth will produce an elevation of the mathematical surface which is expressed by the formula

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1

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where P is the mean density of the earth, & that
of the attracting mass,-the linear unit in expressing A
being a mile. Suppose, for instance, a table-land whose
form is a rectangle of twelve miles by eight miles, having
a height of 500 feet and density half that of the earth; let
the observer be two miles distant from the middle point of
the longer side. The deflection then is 1"-472; but
at one mile it increases to 2"-20. At sixteen astronomical
stations in the English Survey the disturbance of latitude
due to the form of the ground has been computed, and the
At six stations
following will give an idea of the results.
the deflection is under 2", at six others it is between 2"
and 4", and at four stations it exceeds 4". There is one
very exceptional station on the north coast of Banffshire,
near the village of Portsoy, at which the deflection amounts
to 10", so that if that village were placed on a map in a
position to correspond with its astronomical latitude,
it would be 1000 feet out of position! There is the sea to
the north and an undulating country to the south, which,
however, to a spectator at the station does not suggest any
great disturbance of gravity. A somewhat rough estimate
of the local attraction from external causes gives a maximum The deflection at the distance a is
limit of 5", therefore we have 5" unaccounted for, or rather
which must arise from unequal density in the underlying
In order to throw light
strata in the surrounding country.
on this remarkable phenomenon, the latitudes of a number
of stations between Nairn on the west, Fraserburgh on the
east, and the Grampians on the south, were observed, and
the local deflections determined. It is somewhat singular
that the deflections diminish in all directions, not very
regularly certainly, and most slowly in a south-west direc-
tion, finally disappearing, and leaving the maximum at the
original station at Portsoy.

where a is the radius of the (spherical) earth, a(1-7)
the distance of the disturbing mass below the surface, μ
the ratio of the disturbing mass to the mass of the earth,
and a the distance of any point on the surface from that
point, say Q, which is vertically over the disturbing mass.
The maximum value of y is at Q, where it is

A.

k

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uk sin e
(1+k3-2k cos 0)
or since is small, putting h+k=1,.

A

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The maximum deflection takes place at a point whose distance from Q is to the depth of the mass as 1:√ 2, and

its amount is

2 μ 88 እ

If, for instance, the disturbing mass were a sphere a mile in diameter, the excess of its density above that of the surrounding country being equal to half the density of the co-earth, and the depth of its centre half a mile, the greatest deflection would be 5", and the greatest value of y only two inches. Thus a large disturbance of gravity may arise from an irregularity in the mathematical surface whose actual magnitude, as regards height at least, is extremely small.

The method employed by Dr Hutton for computing the attraction of masses of ground is so simple and effectual that it can hardly be improved on. Let a horizontal plane pass through the given station; let r, be the polar ordinates of any point in this plane, and r, 0, z, the coordinates of a particle of the attracting mass; and let it be required to find the attraction of a portion of the mass contained between the horizontal planes z=0, z=h, the cylindrical surfaces r=r1, r=r2, and the vertical planes 0-01, 0-02. The component of the attraction at the station or origin along the line = 0 is

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The effect of the disturbing mass μ on the vibrations of a pendulum would be a maximum at Q; if v be the number of seconds of time gained per diem by the pendulum at Q. and σ the number of seconds of angle in the maximum deflection, then it may be shown that

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so that the number of seconds of time by which at the then on expressing x, 2, 2, ' in terms of u and ", maximum the pendulum is accelerated is about half the number of seconds of angle in the maximum deflection.

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where A (1-e sina ø)}, ▲′-(1 − e2 sin2 p'), and e is the eccentricity. Let f, g, h be the direction cosines of the normal to that plane which contains the normal at P and the point Q, and whose inclinations to the meridian plane of P is a; let also l, m, n and l', m', n' be the direction cosines of the normal at P, and of the tangent to the surface at P which lies in the plane passing through Q, then since the first line is perpendicular to each of the other two and to the chord k, whose direction cosines are proportional to x-x, y-y, z-z, we have these three equations f(x − x)+gy' + h(x − 2) = 0 fl+gm +hn = 0 fl'+gm'+hn' = 0.

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U 1- -cos u cos u' cos - sin u sin u'; also, if v be the third side of a spherical triangle, of which tro sides are u and u' and the included angle, using subsidiary angle such that = e sin "="

sin y sin

COS

2

we obtain finally the following equations:

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These determine rigorously the distance, and the mutual zenith distances and azimuths, of any two points on a spheroid whose latitudes and difference of longitude are given.

By a series of reductions from the equations containing $5 it may be shown that

a+a'−5+ 5' + — — w(p' − p)2 cos ‘q, sin ¢ ̧+ • • • •

neglected. A short computation will show that the small quantity where is the mean of and ', and the higher powers of e are on the right-hand side of this equation can never amount even to the ten thousandth part of a second, which is, practically speaking zero; consequently the sum of the azimuths a+a' on the spheroid equal to the sum of the spherical azimuths, whence follows this very important theorem (known as Dalby's theorem). If, be the latitudes of two points on the surface of a spheroid, their difference of longitude, a, a' their reciprocal azimuths,

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The vertical plane at P passing through Q and the vertical plane at Q passing through P cut the surface of the spheroid in two distinct curves. The greatest distance apart of these curves is, if the mean azimuth of PQ,

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This is a very small quantity; for even in the case of a line of 100 miles in length having a mean azimuth a = 45° in the latitude of Great Britain, it will only amount to half an inch, whilst for a line

of fifty miles it cannot exceed the sixteenth part of an inch. The geodesic line joining P and Q lies wholly between these two curves. If we designate by P', Q' the two curves (the former being that in the vertical plane through P), then, neglecting quantities of the order e203, where @ is the angular distance of P and Q at the centre of the earth, the geodesic curve makes with Pat P an angle equal to the angle it makes with Q'at Q, each of these angles being a third of the angle of intersection of P' and Q'. The difference of length of the geodesic line and either of the curves P, Q' is, s being the length of either,

8 ee cos' po sin 2a. 360

At least this is an approximate expression. Supposing the angle PQ to be as much as 10°, this quantity would be less than oue hundredth of an inch.

An idea of the course of a geodesic line may be gathered from the following example. Let the line be that joining Cadiz and St Petersburg, whose approximate positions are

Lat.

Cadiz.

St Petersburg. 36° 22' N.............59° 56′ N, Long. 6° 18′ w.............30° 17′ E.

If G be the point on the geodesic corresponding to F on that one of the plane curves which contains the normal at Cadiz (by "corresponding" we mean that F and G are on a meridian) then G is to the north of F; at a quarter of the whole distance from Cadiz GF is 458 feet, at half the dis

1 See a paper "On the course of Geodesic Lines on the Earth's Surface" in the Philosophical Magazine for 1870.

tance it is 637 feet, and at three quarters it is 473 feet. The azimuth of the geodesic at Cadiz differs 20" from that of the vertical plane, which is the astronomical azimuth. The azimuth of a geodesic line cannot be observed, so that the line does not enter of necessity into practical geodesy, although many formulæ connected with its use are of great simplicity and elegance. The geodesic line has always held a more important place in the science of geodesy among the mathematicians of France, Germany, and Russia than has been assigned to it in the operations of the English and Indian triangulations. Although the observed angles of a triangulation are not geodesic angles, yet in the calculation of the distance and reciprocal bearings of two points which are far apart, and are connected by a long chain of triangles, we may fall upon the geodesic line in this manner :

...

If A, Z be the points, then to start the calculation from A, we obtain by some preliminary calculation the approximate azimuth of Z, or the angle made by the direction of Z with the side AB or AC of the first triangle. Let P, be the point where this line intersects BC; then, to find P,, where the line cuts the next triangle side CD, we make the angle BP,P, such that BP,P2+BP1A=180°. This fixes P., and P is fixed by a repetition of the same process; so for P4, P5. Now it is clear that the points P1, P2, P3 so computed are those which would be actually fixed by an observer with a theodolite, proceeding in the following manner. Having set the instrument up at A, and turned the telescope in the direction of the computed bearing, an assistant places a mark P1 on the line BC, adjusting it till bisected by the cross-hairs of the telescope at A. The theodolite is then placed over P1, and the telescope turned to A; the horizontal circle is then moved through 180°. The assistant then places a mark P, on the line CD, so as to be bisected by the telescope, which is then moved to P, and in the same manner P, is fixed. Now it is clear that the series of points P1, P2, P3 approaches to the geodesic line, for the plane of any two consecutive elements Pa-1 P, Pa Pa+1 contains the normal at P.. From the formula which we have given above, expressing the mutual relations of two points P, Q on a spheroid, we may obtain the following solution of the problem: Given the latitude of P, with the azimuth a and distance s of Q, to determine the latitude and longitude of Q and the

back azimuth a'.

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sin (--) 2 sin (+0+) a2++∞_cos (K−0−5) ය 2 cos 4(x+0+5) s sin (a+(-a) 02 1+ CoS2 12

8/2

cot

cot

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here p is the radius of curvature of the meridian for the mean latitude (+). These formulæ are approximate only, but they are sufficiently precise even for very long distances.

is

Meridian Arcs.

where

a1 = sin (4,-1) cos (2+1) a= sin 2(-1) cos 2(2+1) a= sin 3(-1) cos 3(2+1) The part of s which depends on n3 is very small; in fact, if we calculate it for the longest arc measured, the Russian arc, it amounts to only an inch and a half, therefore we omit this term, and put for the value

(1 + n + { n3) a − ( 3n+3n3 ) a + (15 n3) a2

Now, if we suppose the observed latitudes to be affected with errors, and that the true latitudes are +, +; and if further we suppose that n+dn is the true value of a-b: a+b, and that n1 itself is merely a very approximate numerical value, we get, on making these substitutions and neglecting the influence of the corrections x on the position of the arc in latitude, i.e., on 、+¢â‚ -(1+n+ n) - (3n, + 3n;) a + (—n;)a,

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here da=x-x; and as b is only known approximately, put b,= (1+u); then we get, after dividing through by the coefficient of dao, which is=1+n1-зn1 cos (2-1) cos (P-1), an equation of the form x=x+h+fu+go, where for convenience we put v for dn. Now in every measured arc there are not only the extremo stations determined in latitude, but also a number of intermediate stations, so that if there be i+1 stations there will be i equatious X2=x1+f2u+g1v + h2

X¡=X2+f¡u+g;v+k.

In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each arc will supply a number of equations in u and v and the corrections to its observed latitudes. Then, according to the method of least squares, those values of u aud v are the most probable which render the sum of the squares of all the errors x a minimum. The corrections which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modern instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections x to the observed latitudes are attributable to local attraction. Archdeacon Pratt, in his treatise on the figure of the earth, objects to this mode of applying least squares first used by Bessel; but certainly Bessel was right, and the objection is groundless.

Comparisons of Standards.

In determining the figure of the earth from the arcs of meridian measured in different countries, one source of uncertainty was, until the last few years, the want of comparisons between the standards of length in which the arcs were expressed. This has been removed by the very

The length of the arc of meridian between the latitudes, and extensive series of comparisons recently made at South

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ampton (see Comparisons of Standard of Length of England, France, Belgium, Prussia, Russia, India, and Australia, made at the Ordnance Survey Office, Southampton, 1866, and a paper in the Philosophical Transac tions for 1873, by Lieut.-Col. A. R. Clarke, C.B., R.E., on the further comparisons of the standards of Austria, Spain, the United States, Cape of Good Hope, and Russia). These direct comparisons, which were carried out with the highest attainable precision, are of very great value. The length of the toise has three independent determinations, viz., through the Russian standard double toise, the Prussian toise, and the Belgium toise,-giving for the length of the toise, expressed in terms of the standard yard of England

through the Russian standard

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.........

6.39453216 ft.

6.39458703 ft. 6.39453215 ft.

Prussian "9 Belgian By combining all the different comparisons made in England and on the Continent on these bars, by the method of least squares, the final value of the toise is

6-39453348 ft. (log = 0.8058088656),

from which the greatest divergence of the three separate results specified above is only half a millionth of a toise, corresponding to ten feet in the earth's radius. From the known ratio of the toise and the metre, 864000 : 443296, we get for the metre

3.28086933 ft. (log = 0.5159889356).

That the close agreement between the determinations of the toise is not due to chance will be seen from the fact that the comparisons of the Prussian toise with the English standard involved 2340 micrometer readings and 520 thermometer readings, extending over twenty-five days, the probable error of the resulting length of the toise being 0.00000015 yard. The probable error of the determination of the Belgian toise is 0.00000027; that of the Russian double toise 000000031. With regard to the metre, there is an independent determination resulting from the comparison of the platinum metre of the Royal Society,a large number of observations giving for the length of the metre 3-28087206 feet, which differs from the former result by about one millionth part. But this determination, involving the expansion of the bar for 30° of temperature, and being dependent on some old observations of Arago, cannot be allowed any weight in modifying the result obtained through the toises. The Russian standard, compared at Southampton, was that on which the length of their base lines and therefore their whole arc depends.

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Dunkirk...

51 2 8.41

Distance of Parallels Feet.

982671-04 988701.92 1657287-93

3710827.13 4509790 84

The latitude of Formentera as here given is taken from the observations of M. Biot, recorded and computed in the third volume of his Traité Élémentaire d'Astronomie physique.

The latitude of the Pantheon, given in the Base du Système Métrique Décimal (ii. 413), 48° 50′ 48"-86. In the Annales de l'Observatoire Imperial de Paris, vol. viii. page 817, we find the latitude of south face of the observatory determined as 48° 50' 11" 71. The Pantheon being 35"-38 north of this, we thus get a second determination of its latitude. The mean is that given above.

The distance of the parallels of Dunkirk and Greenwich, deduced from the recent extension of the triangulation of England into France, in 1862, is 161407 3 feet, which is 3.9 feet greater than that obtained from Captain Kater's triangulation, and 3.2 feet less than the distance calculated by Delambre from General Roy's triangulation. The following table shows the data of the English arc with the distances in standard feet from Formentera.

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The latitude assigned in this table to Saxavord is not the directly observed latitude, which is 60° 49′ 38′′-58, for there astronomically determined; and if we transfer, by means of are here a cluster of three points, whose latitudes are the geodesic connection, the latitude of Gerth of Scare to Saxavord, we get 60° 49′ 36"59; and if we similarly mean of these three is that entered in the above table. transfer the latitude of Bulta, we get 60° 49′ 36"-46. The For the Indian arc in long. 77° 40′ we have the following data:

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Fect

...

1029174-9

12 59 52.165

1756562-0

15 5 53.562

2518376-3

18

3 15.292

8591788-4

21

5 51.532

4697329-5

24 7 11-262

5794695-7

7755835-9

Kaliana.................................. 29 30 48-322

M. Struve's work are as below :—
The data of the Russian arc (long. 26° 40′) taken from

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Feet.

2.94

47 1

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48 45

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50 5 49.95

1737551-48

52 2

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54 39

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56 30 4.97

4076412-28

58 22 47.56

4762421-43

60 5

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62 38

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Having now stated the data of the problem, we may either seek that ellipsoid which best represents the observations, or we may restrict the figure to one of revolution. It will be convenient to commence with the supposition of an ellipsoidal figure, as on so doing we can, by a slight alteration in the equations of minimum, obtain also the required figure of revolution. It may be remarked that, whatever the real figure may be, it is certain that if we presuppose it an ellipsoid, the arithmetical process will bring out an ellipsoid, which ellipsoid will agree better with all the observed latitudes than any spheroid would, therefore we do not prove that it is an ellipsoid; to prove this, arcs of longitude would be required. There is no doubt such arcs will be shortly forthcoming, but as yet they are not available.

The first thing that occurs to one in considering an ellipsoidal earth is the question, What is a meridian curve ? It may be defined in different ways: a point moving on the surface in the direction astronomically determined as "north" might be said to trace a meridian; or we inay define it as the locus of those points which have a constant longitude, whose zeniths lie in a great circle of the heavens, having its poles in the equator; we adopt this definition. Let a, b, c be the semiaxes, c being the polar semiaxis. The equation of the ellipsoid being

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Longitude of a..........
......15° 34' East.

The meridian of the greater axis passes, in the Eastern
Hemisphere, through Spitzbergen, the Straits of Messina,

Hence the equation to a "parallel " in which the latitude Lake Chad in North Africa, and along the west coast of is constant is

IZ.

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cots =0.

So that in an ellipsoidal earth the parallel is no longer a plane curve. Let longitude be reckoned from the plane of As there are two species of latitude, astronomical and geocentric, so there are in the ellipsoidal earth two species of longitude, geocentric (called u) and astronomical (called w). Couceive a line passing through the origin in the plane of thy equator and directed to a point whose longitude is fa+w. The direction cosines of that line are-sin w, cos, aud 0. Those points of the surface whose normals are at right angles to this line are in the meridian whose longitude is w; the condition of perpendicularity is expressed by

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South Africa,-nearly corresponding to the meridian which passes over the greatest quantity of land in that hemisphere. In the Western Hemisphere it passes through Behring's Straits and through the centre of the Pacific the equator passes near North-east Cape on the Arctic Sea, Ocean. The meridian (105° 34′ E.) of the minor axis of through Tong-king and the Straits of Sunda, and corresponds nearly to the meridian which passes over the greatest amount of land in Asia; and in the Western Hemisphere it passes through Smith Sound, the west of Labrador, Montreal, between Cuba and Hayti, and along the west coast of South America, nearly coinciding with the meridian that passes over the greatest amount of land in that hemisphere.

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The length of the meridian quadrant passing through Paris, in the ellipsoidal figure given above, is 10001472.5 metres, showing that the length of the ideal French standard part of the quadrant. The minimum quadrant, in longitude is considerably in error as representing the ten-millionth probable error of the longitude of the major axis of the 105° 34', has a length of 10000024-5 metres. equator given above is of course large, as much perhaps as +15°.

The

It has been objected to this figure of three unequal axes that it does not satisfy, in the proportions of the axes, the conditions brought out in Jacobi's theorem. Admitting this, it has to be noted, on the other hand, that Jacobi's

Take two quantities i, k, such that a2(1−i)—b3⁄41+i)= k2, then theorem contemplates a homogeneous fluid, and this is r2(1-i cos 2u); and take n such that

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=

+ cos 2(u2-up)

k-c k+c 4

k-c

k+c+p cos 2u2+q sin 2u1,

where 4pi cos 2u,, 4q=i sin 2u。.

The normal at P does not pass through the axis of rotation, so that the observed latitudes on an ellipsoid are not exactly the quantities which should be used in the ordinary method of expressing the length of a meridian arc in terms of the latitudes. But it may be shown that this consideration may be neglected.

The data we have collected form 35 equations between the 40 x-corrections to the observed latitudes, and the four unknown quantities determining the elements of the ellipsoid. Suppose n, to be an approximate value of the ratio k-c: k+c, so that

k-c

k+c

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certainly far from the actual condition of our globe, indeed the irregular distribution of continents and oceans suggests as possible a sensible divergence from a perfect surface of revolution.

If we limit the figure to being an ellipsoid of revolution, we get rid in our equations of two unknown quantities, and the result may be expressed thus:

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As might be expected, the sum of the squares of the 40 latitude corrections, viz., 153.99, is greater in this figure than in that of three axes, where it amounts to 138.30. In the Indian arc the largest corrections are at Dodagoontah, +3′′-87, and at Kalianpur, - 3" 68. In the Russian Dic the largest corrections are + 3"-76, at Tornea, and 3" 31. at Staro Nekrassowka. Of the whole 40 corrections, 16 are under 1"0, 10 between 1"0 and 2".0, 10 between 2"-0 and 3"-0, and 4 over 3"0. For the ellipsoidal figure the probable error of an observed latitude is 1"-42; for the spheroidal it would be very slightly larger. This quantity may be taken therefore as approximately the probable amount of local deflection.

In 1860, the Russian Government, at the instance of M. Otto Struve, imperial astronomer at St Petersburg, invited the co-operation of the Governments of Prussia, Belgium, France, and England, to the important end of connecting their respective triangulatious so as to form a continuous chain under the parallel of 52° from the island of Valentia on the south-west coast of Ireland, in longitude 10° 20' 40" W., to Orsk on the river Ural in Russia. This grand undertaking was at once set in action, but up to the present

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