Chapter II: Front Matter (2)

From this it may be shown that the azimuth at A of the geodetic
joining AB is not the same as the astronomical azimuth at A of B or
that determined by the vertical plane A[alpha]B. Generally speaking,
the geodetic lies between the two plane section curves joining A and B
which are formed by the two vertical planes, supposing these points
not far apart. If, however, A and B are nearly in the same latitude,
the geodetic may cross (between A and B) that plane curve which lies
nearest the adjacent pole of the spheroid. The condition of crossing
is this. Suppose that for a moment we drop the consideration of the
earth's non-sphericity, and draw a perpendicular from the pole C on
AB, meeting it in S between A and B. Then A being that point which is
nearest the pole, the geodetic will cross the plane curve if AS be
between 1/4AB and 3/8 AB. If AS lie between this last value and 1/2AB,
the geodetic will lie wholly to the north of both plane curves, that
is, supposing both points to be in the northern hemisphere.

The difference of the azimuths of the vertical section AB and of the
geodetic AB, i.e. the astronomical and geodetic azimuths, is very
small for all observable distances, being approximately:--

Geod. azimuth = Astr. azimuth -(1/12) [e^2/(1 - e^2)] (s^2/[rho]n)
(cos^2[phi] sin 2[alpha] + (s/4a)|sin 2[phi] sin [alpha]), in which: e
and a are the numerical eccentricity and semi-major axis respectively
of the meridian ellipse, [phi] and [alpha] are the latitude and
azimuth at A, s = AB, and [rho] and n are the radii of curvature of
the meridian and perpendicular at A. For s = 100 kilometres, only the
first term is of moment; its value is 0".028 cos^2 [phi] sin 2[alpha],
and it lies well within the errors of observation. If we imagine the
geodetic AB, it will generally trisect the angles between the vertical
sections at A and B, so that the geodetic at A is near the vertical
section AB, and at B near the section BA.[3] The greatest distance of
the vertical sections one from another is e^2s^3 cos^2 [phi]0 sin
2[alpha]0/16a^2, in which [phi]0 and [alpha]0 are the mean latitude
and azimuth respectively of the middle point of AB. For the value s =
64 kilometres, the maximum distance is 3 mm.

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

Cadiz. St Petersburg.
Lat. 36 deg. 22' N. 59 deg. 56' N.
Long. 6 deg. 18' W. 30 deg. 17' E.

If G be the point on the geodetic 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 ft., at half the distance it is 637 ft., and at three-quarters it
is 473 ft. The azimuth of the geodetic at Cadiz differs 20" from that
of the vertical plane, which is the astronomical azimuth.

The azimuth of a geodetic line cannot be observed, so that the line
does not enter of necessity into practical geodesy, although many
formulae connected with its use are of great simplicity and elegance.
The geodetic 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 geodetic 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
geodetic 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 P1 be the point where this line intersects BC; then, to
find P2, where the line cuts the next triangle side CD, we make the
angle BP1P2 such that BP1P2 + BP1A = 180 deg. This fixes P2, and P3 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 deg. The assistant then places a mark P2 on the line CD,
so as to be bisected by the telescope, which is then moved to P2, and
in the same manner P3 is fixed. Now it is clear that the series of
points P1, P2, P3 approaches to the geodetic line, for the plane of
any two consecutive elements P_(n-1) P_n, P_n P_(n+1) contains the
normal at P_n.

If the objection be raised that not the geodetic azimuths but the
astronomical azimuths are observed, it is necessary to consider that
the observed vertical sections do not correspond to points on the
sea-level but to elevated points. Since the normals of the ellipsoid
of rotation do not in general intersect, there consequently arises an
influence of the height on the azimuth. In the case of the measurement
of the azimuth from A to B, the instrument is set to a point A' over
the surface of the ellipsoid (the sea-level), and it is then adjusted
to a point B', also over the surface, say at a height h'. The vertical
plane containing A' and B' also contains A but not B: it must
therefore be rotated through a small azimuth in order to contain B.
The correction amounts approximately to -e^2h' cos^2[phi] sin
2[alpha]/2a; in the case of h' = 1000 m., its value is 0".108
cos^2[phi] sin 2[alpha].

This correction is therefore of greater importance in the case of
observed azimuths and horizontal angles than in the previously
considered case of the astronomical and the geodetic azimuths. The
observed azimuths and horizontal angles must therefore also be
corrected in the case, where it is required to dispense with geodetic
lines.

When the angles of a triangulation have been adjusted by the method of
least squares, and the sides are calculated, 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 position 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 formulae 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 [phi], [phi]' be the latitudes of two
stations A and B; [alpha], [alpha]^* their mutual azimuths counted
from north by east continuously from 0 deg. to 360 deg.; [omega] their
difference of longitude measured from west to east; and s the distance
AB.

First compute a latitude [phi]1 by means of the formula [phi]1 = [phi]
+ (s cos [alpha]) / [rho], where [rho] is the radius of curvature of
the meridian at the latitude [phi]; this will require but four places
of logarithms. Then, in the first two of the following, five places
are sufficient--

s^2 s^2
[epsilon] = ------- sin [alpha] cos a, [eta] = ------- sin^2[alpha] tan[phi]1,
2[rho]n 2[rho]n

s
[phi]' - [phi] = ---- cos ([alpha] - 2/3[epsilon]) - [eta],
rho0

s sin (alpha - 1/3[epsilon])
[omega] = ----------------------------,
n cos ([phi]' + 1/3[eta])

[alpha]^* - [alpha] = [omega] sin ([phi]' + 2/3[eta]) - [epsilon] + 180 deg.

Here n is the normal or radius of curvature perpendicular to the
meridian; both n and [rho] correspond to latitude [phi]1, and [rho]0
to latitude 1/2([phi] + [phi]'). For calculations of latitude and
longitude, tables of the logarithmic values of [rho] sin 1", n sin 1",
and 2 n [rho] sin 1" are necessary. The following table contains these
logarithms for every ten minutes of latitude from 52 deg. to 53 deg.
computed with the elements a = 20926060 and a : b = 295 : 294 :--

+------+------------------+--------------+--------------------+
| | 1 | 1 | 1 |
| Lat. | Log.------------.| Log.--------.| Log.--------------.|
| | [rho] sin 1" | n sin 1" | 2[rho]n sin 1" |
+------+------------------+--------------+--------------------+
|deg. '| | | |
|52 0 | 7.9939434 | 7.9928231 | 0.37131 |
| 10 | 9309 | 8190 | 29 |
| 20 | 9185 | 8148 | 28 |
| 30 | 9060 | 8107 | 26 |
| 40 | 8936 | 8065 | 24 |
| 50 | 8812 | 8024 | 23 |
|53 0 | 8688 | 7982 | 22 |
+------+------------------+--------------+--------------------+

The logarithm in the last column is that required also for the
calculation of spherical excesses, the spherical excess of a triangle
being expressed by a b sin (C/2[rho]n) sin 1".

It is frequently necessary to obtain the co-ordinates of one point
with reference to another point; that is, let a perpendicular arc be
drawn from B to the meridian of A meeting it in P, then, [alpha] being
the azimuth of B at A, the co-ordinates of B with reference to A are

AP = s cos ([alpha] - 2/3[epsilon]), BP = s sin ([alpha] -
1/3[epsilon]),

where [epsilon] is the spherical excess of APB, viz. s^2 sin [alpha]
cos [alpha] multiplied by the quantity whose logarithm is in the
fourth column of the above table.

If it be necessary to determine the geographical latitude and
longitude as well as the azimuths to a greater degree of accuracy than
is given by the above formulae, we make use of the following formula:
given the latitude [phi] of A, and the azimuth [alpha] and the
distance s of B, to determine the latitude [phi]' and longitude
[omega] of B, and the back azimuth [alpha]'. Here it is understood
that [alpha]' is symmetrical to [alpha], so that [alpha]^* + [alpha]'
= 360 deg.

Let

[theta] = s [Delta] / a, where [Delta] = (1 - e^2 sin^2 [phi])^1/2

and

e^2 [theta]^2
[xi] = ------------- cos^2 [phi] sin 2[alpha],
(4 (1 - e^2)

e^2 [theta]^3
[xi]' = ------------- cos^2 [phi] cos^2 [alpha];
(6 (1 - e^2)

[xi], [xi]' are always very minute quantities even for the longest
distances; then, putting [kappa] = 90 deg. - [phi],

[alpha]' + [xi] - [omega] sin 1/2([kappa] - [theta] - [xi]') [alpha]
tan------------------------- = ---------------------------------- cot -------
2 sin 1/2([kappa] + [theta] + [xi]') 2

[alpha]' + [xi] + [omega] cos 1/2([kappa] - [theta] - [xi]') [alpha]
tan------------------------- = ---------------------------------- cot -------
2 cos 1/2([kappa] + [theta] + [xi]') 2

s sin 1/2([alpha]' + [xi] - [alpha]) / [theta]^2 [alpha]' - [alpha]\
[phi]' - [phi] = ----------------------------------------- ( 1 + ---------cos^2 ------------------ );
[rho]0 sin 1/2([alpha]' + [xi] + [alpha]) \ 12 2 /

here [rho]0 is the radius of curvature of the meridian for the mean
latitude 1/2([phi] + [phi]'). These formulae are approximate only, but
they are sufficiently precise even for very long distances.

For lines of any length the formulae of F.W. Bessel (_Astr. Nach._,
1823, iv. 241) are suitable.

If the two points A and B be defined by their geographical
co-ordinates, we can accurately calculate the corresponding
astronomical azimuths, i.e. those of the vertical section, and then
proceed, in the case of not too great distances, to determine the
length and the azimuth of the shortest lines. For _any_ distances
recourse must again be made to Bessel's formula.[4]

Let [alpha], [alpha]' be the mutual azimuths of two points A, B on a
spheroid, k the chord line joining them, [mu], [mu]' the angles made
by the chord with the normals at A and B, [phi], [phi]', [omega] their
latitudes and difference of longitude, and (x^2 + y^2)/a^2 + z^2 b^2 =
1 the equation of the surface; then if the plane xz passes through A
the co-ordinates of A and B will be

x = (a/[Delta]) cos [phi], x' = (a/[Delta]') cos [phi]' cos [omega],

y = 0 y' = (a/[Delta]') cos [phi]' sin [omega],

z = (a/[Delta]) (1 - e^2) sin [phi], z' = (a/[Delta]') (1 - e^2) sin [phi]',

where [Delta] = (1 - e^2 sin^2 [phi])^1/2, [Delta]' = (1 - e^2 sin^2
[phi]')^1/2, and e is the eccentricity. Let f, g, h be the direction
cosines of the normal to that plane which contains the normal at A and
the point B, and whose inclinations to the meridian plane of A is =
[alpha]; let also l, m, n and l', m', n' be the direction cosines of
the normal at A, and of the tangent to the surface at A which lies in
the plane passing through B, 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(z' - z) = 0

fl + gm + hn = 0

fl' + gm' + hn' = 0.

Eliminate f, g, h from these equations, and substitute

l = cos [phi] l' = - sin [phi] cos [alpha]

m = 0 m' = sin [alpha]

n = sin [phi] n' = cos [phi] cos [alpha],

and we get

(x' - x) sin [phi] + y' cot [alpha] - (z' - z) cos [phi] = 0.

The substitution of the values of x, z, x', y', z' in this equation
will give immediately the value of cot [alpha]; and if we put [zeta],
[zeta]' for the corresponding azimuths on a sphere, or on the
supposition e = 0, the following relations exist

cos [phi] Q
cot [alpha] - cot [zeta] = e^2 ------------------
cos [phi]' [Delta]

cos [phi]' Q
cot [alpha]' - cot [zeta]' = e^2 ------------------
cos [phi] [Delta]'

[Delta]' sin [phi] - [Delta] sin [phi]' = Q sin [omega].

If from B we let fall a perpendicular on the meridian plane of A, and
from A let fall a perpendicular on the meridian plane of B, then the
following equations become geometrically evident:

k sin [mu] sin [alpha] = (a/[Delta]') cos [phi]' sin [omega]

k sin [mu]' sin [alpha]' = (a/[Delta]) cos [phi] sin [omega].

Now in any surface u = 0 we have

k^2 = (x' - x)^2 + (y' - y)^2 + (z' - z)^2
_ _
| du du du | / / du^2 du^2 du^2 \ 1/2
-cos [mu] = |(x' - x) -- + (y' - y) -- + (z' - z) -- | / k ( ---- + ---- + ---- )
|_ dx dy dz_|/ \ dx^2 dy^2 dz^2 /
_ _
| du du du | / / du^2 du^2 du^2 \ 1/2
-cos [mu]' = |(x' - x) --- + (y' - y) --- + (z' - z) --- | / k ( ----- + ----- + ----- ).
|_ dx' dy' dz'_|/ \ dx'^2 dy'^2 dz'^2 /

In the present case, if we put

xx' zz'
1 - --- - --- = U,
a^2 b^2

then

k^2 /z' - z \ ^2
--- = 2U - e^2 ( ------ )
a^2 \ b /

cos [mu] = (a/k) [Delta]U; cos [mu]' = (a/k) [Delta]'U.

Let u be such an angle that

(1 - e^2)^1/2 sin [phi] = [Delta] sin u

cos [phi] = [Delta] cos u,

then on expressing x, x', z, z' in terms of u and u',

U = 1 - cos u cos u' cos [omega] - sin u sin u';

also, if v be the third side of a spherical triangle, of which two
sides are 1/2[pi] - u and 1/2[pi] - u' and the included angle [omega],
using a subsidiary angle [psi] such that

sin [psi] sin 1/2v = e sin 1/2(u' - u) cos 1/2(u' + u),

we obtain finally the following equations:--

k = 2a cos [psi] sin 1/2v

cos [mu] = [Delta] sec [psi] sin 1/2v

cos [mu]' = [Delta]' sec [psi] sin 1/2v

sin [mu] sin [alpha] = (a/k) cos u' sin [omega]

sin [mu]' sin [alpha]' = (a/k) cos u sin [omega].

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 [zeta],
[zeta]' it may be shown that

[alpha] + [alpha]' = [zeta] + [zeta]' + 1/4e^4[omega]([phi]' - [phi])^2
cos^4 [phi]0 sin [phi]0 + ...,

where [phi]0 is the mean of [phi] and [phi]', and the higher powers of
e are neglected. A short computation will show that the small quantity
on the right-hand side of this equation cannot amount even to the
thousandth part of a second for k < 0.1a, which is, practically
speaking, zero; consequently the sum of the azimuths [alpha] +
[alpha]' on the spheroid is equal to the sum of the spherical
azimuths, whence follows this very important theorem (known as Dalby's
theorem). If [phi], [phi]' be the latitudes of two points on the
surface of a spheroid, [omega] their difference of longitude, [alpha],
[alpha]' their reciprocal azimuths,

tan 1/2[omega] = cot 1/2([alpha] + [alpha]') {cos 1/2([phi]' - [phi])/
sin 1/2([phi]' + [phi])}.

The computation of the geodetic from the astronomical azimuths has
been given above. From k we can now compute the length s of the
vertical section, and from this the shortest length. The difference of
length of the geodetic line and either of the plane curves is

e^4 s^5 cos^4 [phi]0 sin^2 2[alpha]0/360 a^4.

At least this is an approximate expression. Supposing s = 0.1a, this
quantity would be less than one-hundredth of a millimetre. The line s
is now to be calculated as a circular arc with a mean radius r along
AB. If [phi]0 = 1/2([phi] + [phi]'), [alpha]0 = 1/2(180 deg. + [alpha]
- [alpha]'), [Delta]0 = (1 - e^2 sin^2 [phi]0)^1/2, then 1/r =
[Delta]0/a [1 + e^2/(1 - e^2) (cos^2 [phi]0 cos^2 [alpha]0)], and
approximately sin (s/2r) = k/2r. These formulae give, in the case of k
= 0.1a, values certain to eight logarithmic decimal places. An
excellent series of formulae for the solution of the problem, to
determine the azimuths, chord and distance along the surface from the
geographical co-ordinates, was given in 1882 by Ch. M. Schols
(_Archives Neerlandaises_, vol. xvii.).

_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 eccentricity. In
the first place, supposing the earth a sphere covered with a film of
water, let the density [rho] 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 [rho], a function of r only,
but is expressed by [rho] + [rho]', where [rho]' is a function of
three co-ordinates [theta], [phi], r. Then [rho]' 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 [rho]' 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' = N. The knowledge of N 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 [rho]', M the mass of the earth (the
attraction-constant is assumed equal to unity)

M M M
----- + V = C = -- - --- N + V.
a + N a a^2

As far as we know, N is always a very small quantity, and we have with
sufficient approximation N = 3V/4[pi][delta]a, where [delta] is the
mean density of the earth. Thus we have the disturbance in elevation
of the sea-level expressed in terms of the potential of the disturbing
matter. If at any point P the value of N remain constant when we pass
to any adjacent point, then the actual surface is there parallel to
the ideal spherical surface; as a rule, however, the normal at P is
inclined to that at P', and astronomical observations have shown that
this inclination, the deflection or deviation, amounting ordinarily to
one or two seconds, may in some cases exceed 10", or, as at the foot
of the Himalayas, even 60". 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 should 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
Unregelmassigkeiten der Figur der Erde auf geodatische 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.

Let there be at the station an attraction to the north-east throwing
the zenith to the south-west, so that it takes in the celestial sphere
a position Z', its undisturbed position being Z. Let the rectangular
components of the displacement ZZ' be [xi] measured southwards and
[eta] measured westwards. Now the great circle joining Z' with the
pole of the heavens P makes there an angle with the meridian PZ =
[eta] cosec PZ' = [eta] sec [phi], where [phi] is the latitude of the
station. Also this great circle meets the horizon in a point whose
distance from the great circle PZ is [eta] sec [phi] sin [phi] = [eta]
tan [phi]. 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 [xi]; the observed longitude
a correction [eta] sec [phi]; and any observed azimuth a correction
[eta] tan [phi]. Here it is supposed that azimuths are measured from
north by east, and longitudes eastwards. The horizontal angles are
also influenced by the deflections of the plumb-line, in fact, just as
if the direction of the vertical axis of the theodolite varied by the
same amount. This influence, however, is slight, so long as the sights
point almost horizontally at the objects, which is always the case in
the observation of distant points.

The expression given for N enables one to form an approximate estimate
of the effect of a compact mountain in raising 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 in. 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 we may ascertain that an
elevation amounting to several hundred feet may exist near their base.
The geodetical operations, however, rather negative this idea, for it
was shown by Colonel Clarke (_Phil. Mag._, 1878) that the form of the
sea-level along the Indian arc departs but slightly from that of the
mean figure of the earth. If this be so, the action of the Himalayas
must be counteracted by subterranean tenuity.

Suppose now that A, B, C, ... are the stations of a network of
triangulation projected on or lying on a spheroid of semiaxis major
and eccentricity 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
reference 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 [xi] and to the east by the quantity [eta]. Then in
starting the calculation of geodetic latitudes, longitudes and
azimuths from A, we must take, not the observed elements [phi],
[alpha], but for [phi], [phi] + [xi], and for [alpha], [alpha] + [eta]
tan [phi], and zero longitude must be replaced by [eta] sec [phi]. 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 ([phi]'), ([alpha]'), ([omega]) 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

([phi]') + f[xi] + g[eta] + hda + kde,

([alpha]') + f'[xi] + g'[eta] + h'da + k'de,

[omega] + f"[xi] + g"[eta] + 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 [xi]', [eta]' be the components of the inclination
at that point, then we have

[xi]' = ([phi]') - [phi]' + f[xi] + g[eta] + hda + kde,

[eta]' tan [phi]' = ([alpha]') - [alpha]' + f'[xi] + g'[eta] + h'da + k'de,

[eta]' sec [phi]' = ([omega]) - [omega] + f"[xi] + g"[eta] + h"da + k"de,

where [phi]', [alpha]', [omega] 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.

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 [xi], [eta], da, de,
which make the quantity [xi]^2 + [eta]^2 + [xi]'^2 + [eta]'^2 + ... a
minimum. Thus we obtain that spheroid which best represents the
surface covered by the triangulation.

In the _Account of the Principal Triangulation of Great Britain and
Ireland_ will be found the determination, from 75 equations, of the
spheroid best representing the surface of the British Isles. Its
elements are a = 20927005 [+-] 295 ft., b : a - b = 280 [+-] 8; and it
is so placed that at Greenwich Observatory [xi] = 1".864, [eta] =
-0".546.

Taking Durham Observatory as the origin, and the tangent plane to the
surface (determined by [xi] = -0".664, [eta] = -4".117) as the plane
of x and y, the former measured northwards, and z measured vertically
downwards, the equation to the surface is

.99524953 x^2 + .99288005 y^2 + .99763052 z^2 - 0.00671003xz - 41655070z = 0.

_Altitudes._

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. Of great
importance is a knowledge of the height of the base for its reduction
to the sea-level. Again the height of a station does influence a
little the observation of terrestrial angles, for a vertical line at B
does not lie generally in the vertical plane of A (see above). The
height above the sea-level also influences the geographical latitude,
inasmuch as the centrifugal force is increased and the magnitude and
direction of the attraction of the earth are altered, and the effect
upon the latitude is a very small term expressed by the formula h (g'-
g) sin 2 [phi] / ag, where g, g' are the values of gravity at the
equator and at the pole. This is h sin 2 [phi] / 5820 seconds, h being
in metres, a quantity which may be neglected, since for ordinary
mountain heights it amounts to only a few hundredths of a second. We
can assume this amount as joined with the northern component of the
plumb-line perturbations.

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 .0792 [+-] .0047; 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
deg. + [delta], 90 deg. + [delta]' their mutual zenith distances
([delta] being that observed at h), s their distance apart, the earth
being regarded as a sphere of radius = a, then, with sufficient
precision,

/ 1 - 2k \ / 1 - 2k \
h' - h = s tan ( s -------- - [delta] ), h - h' = s tan ( -------- - [delta]' ).
\ 2a / \ 2a /

If from a station whose height is h the horizon of the sea be observed
to have a zenith distance 90 deg. + [delta], then the above formula
gives for h the value

a tan^2 [delta]
h = -- -------------.
2 1 - 2k

Suppose the depression [delta] to be n minutes, then h = 1.054n^2 if
the ray be for the greater part of its length crossing the sea; if
otherwise, h = 1.040n^2. 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 deg. 4' 48", or [delta] = 64.8; the ray is pretty
equally disposed over land and water, and hence h = 1.047n^2 = 4396
ft. The actual height of the hill by spirit-levelling is 4406 ft., so
that the error of the height thus obtained is only 10 ft.

The determination of altitudes by means of spirit-levelling is
undoubtedly the most exact method, particularly in its present
development as precise-levelling, by which there have been determined
in all civilized countries close-meshed nets of elevated points
covering the entire land. (A. R. C; F. R. H.)

FOOTNOTES:

[1] An arrangement acting similarly had been previously introduced by
Borda.

[2] _Geodetic Survey of South Africa_, vol. iii. (1905), p. viii;
_Les Nouveaux Appareils pour la mesure rapide des bases geod._, par
J. Rene Benoit et Ch. Ed. Guillaume (1906).

[3] See a paper "On the Course of Geodetic Lines on the Earth's
Surface" in the _Phil. Mag._ 1870; Helmert, _Theorien der hoheren
Geodasie_, 1. 321.

[4] Helmert, Theorien der hoheren Geodasie, 1. 232, 247.

GEOFFREY, surnamed MARTEL (1006-1060), count of Anjou, son of the count Fulk Nerra (q.v.) and of the countess Hildegarde or Audegarde, was born on the 14th of October 1006. During his father's lifetime he was recognized as suzerain by Fulk l'Oison ("the Gosling"), count of Vendome, the son of his half-sister Adela. Fulk having revolted, he confiscated the countship, which he did not restore till 1050. On the 1st of January 1032 he married Agnes, widow of William the Great, duke of Aquitaine, and taking arms against William the Fat, eldest son and successor of William the Great, defeated him and took him prisoner at Mont-Couer near Saint-Jouin-de-Marnes on the 20th of September 1033. He then tried to win recognition as dukes of Aquitaine for the sons of his wife Agnes by William the Great, who were still minors, but Fulk Nerra promptly took up arms to defend his suzerain William the Fat, from whom he held the Loudunois and Saintonge in fief against his son. In 1036 Geoffrey Martel had to liberate William the Fat, on payment of a heavy ransom, but the latter having died in 1038, and the second son of William the Great, Odo, duke of Gascony, having fallen in his turn at the siege of Mauze (10th of March 1039) Geoffrey made peace with his father in the autumn of 1039, and had his wife's two sons recognized as dukes. About this time, also, he had interfered in the affairs of Maine, though without much result, for having sided against Gervais, bishop of Le Mans, who was trying to make himself guardian of the young count of Maine, Hugh, he had been beaten and forced to make terms with Gervais in 1038. In 1040 he succeeded his father in Anjou and was able to conquer Touraine (1044) and assert his authority over Maine (see ANJOU). About 1050 he repudiated Agnes, his first wife, and married Grecie, the widow of Bellay, lord of Montreuil-Bellay (before August 1052), whom he subsequently left in order to marry Adela, daughter of a certain Count Odo. Later he returned to Grecie, but again left her to marry Adelaide the German. When, however, he died on the 14th of November 1060, at the monastery of St Nicholas at Angers, he left no children, and transmitted the countship to Geoffrey the Bearded, the eldest of his nephews (see ANJOU).

See Louis Halphen, _Le Comte d'Anjou au XI^e siecle_ (Paris, 1906). A
summary biography is given by Celestin Port, _Dictionnaire historique,
geographique et biographique de Maine-et-Loire_ (3 vols.,
Paris-Angers, 1874-1878), vol. ii. pp. 252-253, and a sketch of the
wars by Kate Norgate, _England under the Angevin Kings_ (2 vols.,
London, 1887), vol. i. chs. iii. iv. (L. H.*)

GEOFFREY, surnamed PLANTAGENET [or PLANTEGENET] (1113-1151), count of Anjou, was the son of Count Fulk the Young and of Eremburge (or Arembourg of La Fleche); he was born on the 24th of August 1113. He is also called "le bel" or "the handsome," and received the surname of Plantagenet from the habit which he is said to have had of wearing in his cap a sprig of broom (_genet_). In 1127 he was made a knight, and on the 2nd of June 1129 married Matilda, daughter of Henry I. of England, and widow of the emperor Henry V. Some months afterwards he succeeded to his father, who gave up the countship when he definitively went to the kingdom of Jerusalem. The years of his government were spent in subduing the Angevin barons and in conquering Normandy (see ANJOU). In 1151, while returning from the siege of Montreuil-Bellay, he took cold, in consequence of bathing in the Loir at Chateau-du-Loir, and died on the 7th of September. He was buried in the cathedral of Le Mans. By his wife Matilda he had three sons: Henry Plantagenet, born at Le Mans on Sunday, the 5th of March 1133; Geoffrey, born at Argentan on the 1st of June 1134; and William Long-Sword, born on the 22nd of July 1136.

See Kate Norgate, _England under the Angevin Kings_ (2 vols., London,
1887), vol. i. chs. v.-viii.; Celestin Port, _Dictionnaire historique,
geographique et biographique de Maine-et-Loire_ (3 vols.,
Paris-Angers, 1874-1878), vol. ii. pp. 254-256. A history of Geoffrey
le Bel has yet to be written; there is a biography of him written in
the 12th century by Jean, a monk of Marmoutier, _Historia Gaufredi,
ducis Normannorum et comitis Andegavorum_, published by Marchegay et
Salmon; "Chroniques des comtes d'Anjou" (_Societe de l'histoire de
France_, Paris, 1856), pp. 229-310. (L. H.*)

GEOFFREY (1158-1186), duke of Brittany, fourth son of the English king Henry II. and his wife Eleanor of Aquitaine, was born on the 23rd of September 1158. In 1167 Henry suggested a marriage between Geoffrey and Constance (d. 1201), daughter and heiress of Conan IV., duke of Brittany (d. 1171); and Conan not only assented, perhaps under compulsion, to this proposal, but surrendered the greater part of his unruly duchy to the English king. Having received the homage of the Breton nobles, Geoffrey joined his brothers, Henry and Richard, who, in alliance with Louis VII. of France, were in revolt against their father; but he made his peace in 1174, afterwards helping to restore order in Brittany and Normandy, and aiding the new French king, Philip Augustus, to crush some rebellious vassals. In July 1181 his marriage with Constance was celebrated, and practically the whole of his subsequent life was spent in warfare with his brother Richard. In 1183 he made peace with his father, who had come to Richard's assistance; but a fresh struggle soon broke out for the possession of Anjou, and Geoffrey was in Paris treating for aid with Philip Augustus, when he died on the 19th of August 1186. He left a daughter, Eleanor, and his wife bore a posthumous son, the unfortunate Arthur.

GEOFFREY (c. 1152-1212), archbishop of York, was a bastard son of Henry II., king of England. He was distinguished from his legitimate half-brothers by his consistent attachment and fidelity to his father. He was made bishop of Lincoln at the age of twenty-one (1173); but though he enjoyed the temporalities he was never consecrated and resigned the see in 1183. He then became his father's chancellor, holding a large number of lucrative benefices in plurality. Richard nominated him archbishop of York in 1189, but he was not consecrated till 1191, or enthroned till 1194. Geoffrey, though of high character, was a man of uneven temper; his history in chiefly one of quarrels, with the see of Canterbury, with the chancellor William Longchamp, with his half-brothers Richard and John, and especially with his canons at York. This last dispute kept him in litigation before Richard and the pope for many years. He led the clergy in their refusal to be taxed by John and was forced to fly the kingdom in 1207. He died in Normandy on the 12th of December 1212.

See Giraldus Cambrensis, _Vita Galfridi_; Stubbs's prefaces to _Roger
de Hoveden_, vols. iii. and iv. (Rolls Series). (H. W. C. D.)

GEOFFREY DE MONTBRAY (d. 1093), bishop of Coutances (_Constantiensis_), a right-hand man of William the Conqueror, was a type of the great feudal prelate, warrior and administrator at need. He knew, says Orderic, more about marshalling mailed knights than edifying psalm-singing clerks. Obtaining, as a young man, in 1048, the see of Coutances, by his brother's influence (see MOWBRAY), he raised from his fellow nobles and from their Sicilian spoils funds for completing his cathedral, which was consecrated in 1056. With bishop Odo, a warrior like himself, he was on the battle-field of Hastings, exhorting the Normans to victory; and at William's coronation it was he who called on them to acclaim their duke as king. His reward in England was a mighty fief scattered over twelve counties. He accompanied William on his visit to Normandy (1067), but, returning, led a royal force to the relief of Montacute in September 1069. In 1075 he again took the field, leading with Bishop Odo a vast host against the rebel earl of Norfolk, whose stronghold at Norwich they besieged and captured.

Meanwhile the Conqueror had invested him with important judicial functions. In 1072 he had presided over the great Kentish suit between the primate and Bishop Odo, and about the same time over those between the abbot of Ely and his despoilers, and between the bishop of Worcester and the abbot of Ely, and there is some reason to think that he acted as a Domesday commissioner (1086), and was placed about the same time in charge of Northumberland. The bishop, who attended the Conqueror's funeral, joined in the great rising against William Rufus next year (1088), making Bristol, with which (as Domesday shows) he was closely connected and where he had built a strong castle, his base of operations. He burned Bath and ravaged Somerset, but had submitted to the king before the end of the year. He appears to have been at Dover with William in January 1090, but, withdrawing to Normandy, died at Coutances three years later. In his fidelity to Duke Robert he seems to have there held out for him against his brother Henry, when the latter obtained the Cotentin.

See E.A. Freeman, _Norman Conquest_ and _William Rufus_; J.H. Round,
_Feudal England_; and, for original authorities, the works of Orderic
Vitalis and William of Poitiers, and of Florence of Worcester; the
Anglo-Saxon Chronicle; William of Malmesbury's _Gesta pontificum_, and
Lanfranc's works, ed. Giles; Domesday Book. (J. H. R.)

GEOFFREY OF MONMOUTH (d. 1154), bishop of St Asaph and writer on early British history, was born about the year 1100. Of his early life little is known, except that he received a liberal education under the eye of his paternal uncle, Uchtryd, who was at that time archdeacon, and subsequently bishop, of Llandaff. In 1129 Geoffrey appears at Oxford among the witnesses of an Oseney charter. He subscribes himself Geoffrey Arturus; from this we may perhaps infer that he had already begun his experiments in the manufacture of Celtic mythology. A first edition of his _Historia Britonum_ was in circulation by the year 1139, although the text which we possess appears to date from 1147. This famous work, which the author has the audacity to place on the same level with the histories of William of Malmesbury and Henry of Huntingdon, professes to be a translation from a Celtic source; "a very old book in the British tongue" which Walter, archdeacon of Oxford, had brought from Brittany. Walter the archdeacon is a historical personage; whether his book has any real existence may be fairly questioned. There is nothing in the matter or the style of the _Historia_ to preclude us from supposing that Geoffrey drew partly upon confused traditions, partly on his own powers of invention, and to a very slight degree upon the accepted authorities for early British history. His chronology is fantastic and incredible; William of Newburgh justly remarks that, if we accepted the events which Geoffrey relates, we should have to suppose that they had happened in another world. William of Newburgh wrote, however, in the reign of Richard I. when the reputation of Geoffrey's work was too well established to be shaken by such criticisms. The fearless romancer had achieved an immediate success. He was patronized by Robert, earl of Gloucester, and by two bishops of Lincoln; he obtained, about 1140, the archdeaconry of Llandaff "on account of his learning"; and in 1151 was promoted to the see of St Asaph.

Before his death the _Historia Britonum_ had already become a model and a quarry for poets and chroniclers. The list of imitators begins with Geoffrey Gaimar, the author of the _Estorie des Engles_ (c. 1147), and Wace, whose _Roman de Brut_ (1155) is partly a translation and partly a free paraphrase of the _Historia_. In the next century the influence of Geoffrey is unmistakably attested by the _Brut_ of Layamon, and the rhyming English chronicle of Robert of Gloucester. Among later historians who were deceived by the _Historia Britonum_ it is only needful to mention Higdon, Hardyng, Fabyan (1512), Holinshed (1580) and John Milton. Still greater was the influence of Geoffrey upon those writers who, like Warner in _Albion's England_ (1586), and Drayton in _Polyolbion_ (1613), deliberately made their accounts of English history as poetical as possible. The stories which Geoffrey preserved or invented were not infrequently a source of inspiration to literary artists. The earliest English tragedy, _Gorboduc_ (1565), the _Mirror for Magistrates_ (1587), and Shakespeare's Lear, are instances in point. It was, however, the Arthurian legend which of all his fabrications attained the greatest vogue. In the work of expanding and elaborating this theme the successors of Geoffrey went as far beyond him as he had gone beyond Nennius; but he retains the credit due to the founder of a great school. Marie de France, who wrote at the court of Henry II., and Chretien de Troyes, her French contemporary, were the earliest of the avowed romancers to take up the theme. The succeeding age saw the Arthurian story popularized, through translations of the French romances, as far afield as Germany and Scandinavia. It produced in England the _Roman du Saint Graal_ and the _Roman de Merlin_, both from the pen of Robert de Borron; the _Roman de Lancelot_; the _Roman de Tristan_, which is attributed to a fictitious Lucas de Gast. In the reign of Edward IV. Sir Thomas Malory paraphrased and arranged the best episodes of these romances in English prose. His _Morte d'Arthur_, printed by Caxton in 1485, epitomizes the rich mythology which Geoffrey's work had first called into life, and gave the Arthurian story a lasting place in the English imagination. The influence of the _Historia Britonum_ may be illustrated in another way, by enumerating the more familiar of the legends to which it first gave popularity. Of the twelve books into which it is divided only three (Bks. IX., X., XI.) are concerned with Arthur. Earlier in the work, however, we have the adventures of Brutus; of his follower Corineus, the vanquisher of the Cornish giant Goemagol (Gogmagog); of Locrinus and his daughter Sabre (immortalized in Milton's _Comus_); of Bladud the builder of Bath; of Lear and his daughters; of the three pairs of brothers, Ferrex and Porrex, Brennius and Belinus, Elidure and Peridure. The story of Vortigern and Rowena takes its final form in the _Historia Britonum_; and Merlin makes his first appearance in the prelude to the Arthur legend. Besides the _Historia Britonum_ Geoffrey is also credited with a _Life of Merlin_ composed in Latin verse. The authorship of this work has, however, been disputed, on the ground that the style is distinctly superior to that of the _Historia_. A minor composition, the _Prophecies of Merlin_, was written before 1136, and afterwards incorporated with the _Historia_, of which it forms the seventh book.

For a discussion of the manuscripts of Geoffrey's work, see Sir T.D.
Hardy's _Descriptive Catalogue_ (Rolls Series), i. pp. 341 ff. The
_Historia Britonum_ has been critically edited by San Marte (Halle,
1854). There is an English translation by J.A. Giles (London, 1842).
The _Vita Merlini_ has been edited by F. Michel and T. Wright (Paris,
1837). See also the _Dublin Univ. Magazine_ for April 1876, for an
article by T. Gilray on the literary influence of Geoffrey; G.
Heeger's _Trojanersage der Britten_ (1889); and La Borderie's _Etudes
historiques bretonnes_ (1883). (H. W. C. D.)

GEOFFREY OF PARIS (d. c. 1320), French chronicler, was probably the author of the _Chronique metrique de Philippe le Bel, or Chronique rimee de Geoffroi de Paris_. This work, which deals with the history of France from 1300 to 1316, contains 7918 verses, and is valuable as that of a writer who had a personal knowledge of many of the events which he relates. Various short historical poems have also been attributed to Geoffrey, but there is no certain information about either his life or his writings.

The _Chronique_ was published by J.A. Buchon in his _Collection des
chroniques_, tome ix. (Paris, 1827), and it has also been printed in
tome xxii. of the _Recueil des historiens des Gaules et de la France_
(Paris, 1865). See G. Paris, _Histoire de la litterature francaise au
moyen age_ (Paris, 1890); and A. Molinier, _Les Sources de l'histoire
de France_, tome iii. (Paris, 1903).

GEOFFREY THE BAKER (d. c. 1360), English chronicler, is also called Walter of Swinbroke, and was probably a secular clerk at Swinbrook in Oxfordshire. He wrote a _Chronicon Angliae temporibus Edwardi II. et Edwardi III._, which deals with the history of England from 1303 to 1356. From the beginning until about 1324 this work is based upon Adam Murimuth's _Continuatio chronicarum_, but after this date it is valuable and interesting, containing information not found elsewhere, and closing with a good account of the battle of Poitiers. The author obtained his knowledge about the last days of Edward II. from William Bisschop, a companion of the king's murderers, Thomas Gurney and John Maltravers. Geoffrey also wrote a _Chroniculum_ from the creation of the world until 1336, the value of which is very slight. His writings have been edited with notes by Sir E.M. Thompson as the _Chronicon Galfridi le Baker de Swynebroke_ (Oxford, 1889). Some doubt exists concerning Geoffrey's share in the compilation of the _Vita et mors Edwardi II._, usually attributed to Sir Thomas de la More, or Moor, and printed by Camden in his _Anglica scripta_. It has been maintained by Camden and others that More wrote an account of Edward's reign in French, and that this was translated into Latin by Geoffrey and used by him in compiling his _Chronicon_. Recent scholarship, however, asserts that More was no writer, and that the _Vita et mors_ is an extract from Geoffrey's _Chronicon_, and was attributed to More, who was the author's patron. In the main this conclusion substantiates the verdict of Stubbs, who has published the _Vita et mors_ in his _Chronicles of the reigns of Edward I. and Edward II._ (London, 1883). The manuscripts of Geoffrey's works are in the Bodleian library at Oxford.