Chapter I: Front Matter (1)

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Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
paragraphs.

(4) Macrons and breves above letters and dots below letters were not
inserted.

(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
letters.

(6) The following typographical errors have been corrected:

ARTICLE GEOFFREY: "... 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." 'William' amended from 'Willian'.

ARTICLE GEOLOGY: "... and at the same time greater appreciation has
been shown of the signification and strength of the geological
proofs of the high antiquity of our planet." 'strength' amended
from 'stength'.

ARTICLE GEOLOGY: "... it can be demonstrated that sometimes an inch
or two of sediment might, on certain horizons, represent the
deposit of an enormously longer period than a hundred or a thousand
times the same amount of sediment on other horizons." 'might'
amended from 'much'.

ARTICLE GEOLOGY: "But no such extensive disturbance of the normal
conditions of the distribution of life can take place without
carrying with it many secondary effects, and setting in motion a
wide cycle of change and of reaction in the animal and vegetable
kingdoms." 'kingdoms' amended from 'kindgoms'.

ARTICLE GEOMETRY: "The bases and altitudes of equal solid
parallelepipeds are reciprocally proportional; and if the bases and
altitudes be reciprocally proportional, the solid parallelepipeds
are equal." 'are' amended from 'and'.

ARTICLE GEOMETRY: "An alternative method of testing a relation is
illustrated in the following example:--If A, B, C, D, E,
F be six collinear points, then" 'following example:--' amended
from 'example: following'.

ARTICLE GEOMETRY: "3. In an hyperbolic involution any two conjugate
points are harmonic conjugates with regard to the two foci." 'an'
amended from 'a'.

ARTICLE GEOMETRY: "If two lines, given by their projections,
intersect, the intersection of their planes and the intersection of
their elevations must lie in a line perpendicular to the axis,
because they must be the projections of the point common to the two
lines." 'planes' amended from 'plans'.

ARTICLE GEOMETRY: "Where this is the case, if [alpha] be the measure
of curvature, the linear element can be put into the form" 'if'
amended from 'it'.

ARTICLE GEOMETRY: "The development of the consequences of these
metrical definitions is the subject of non-Euclidean geometry."
'subject' amended from 'subjct'.

ENCYCLOPAEDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME XI, SLICE VI

GEODESY to GEOMETRY

ARTICLES IN THIS SLICE:

GEODESY GEOFFROY, ETIENNE FRANCOIS
GEOFFREY (Martel) GEOFFROY, JULIEN LOUIS
GEOFFREY (Plantagenet) GEOFFROY SAINT-HILAIRE, ETIENNE
GEOFFREY (duke of Brittany) GEOFFROY SAINT-HILAIRE, ISIDORE
GEOFFREY (archbishop of York) GEOGRAPHY
GEOFFREY DE MONTBRAY GEOID
GEOFFREY OF MONMOUTH GEOK-TEPE
GEOFFREY OF PARIS GEOLOGY
GEOFFREY THE BAKER GEOMETRICAL CONTINUITY
GEOFFRIN, MARIE THERESE RODET GEOMETRY

GEODESY (from the Gr. [Greek: ge], the earth, and [Greek: daiein], to divide), the science of surveying (q.v.) extended to large tracts of country, having in view not only the production of a system of maps of very great accuracy, but the determination of the curvature of the surface of the earth, and eventually of the figure and dimensions of the earth. This last, indeed, may be the sole object in view, as was the case in the operations conducted in Peru and in Lapland by the celebrated French astronomers P. Bouguer, C.M. de la Condamine, P.L.M. de Maupertuis, A.C. Clairault and others; and the measurement of the meridian arc of France by P.F.A. Mechain and J.B.J. Delambre had for its end the determination of the true length of the "metre" which was to be the legal standard of length of France (see EARTH, FIGURE OF THE).

The basis of every extensive survey is an accurate triangulation, and the operations of geodesy consist in the measurement, by theodolites, of the angles of the triangles; the measurement of one or more sides of these triangles on the ground; the determination by astronomical observations of the azimuth of the whole network of triangles; the determination of the actual position of the same on the surface of the earth by observations, first for latitude at some of the stations, and secondly for longitude; the determination of altitude for all stations.

For the computation, the points of the actual surface of the earth are imagined as projected along their plumb lines on the mathematical figure, which is given by the stationary sea-level, and the extension of the sea through the continents by a system of imaginary canals. For many purposes the mathematical surface is assumed to be a plane; in other cases a sphere of radius 6371 kilometres (20,900,000 ft.). In the case of extensive operations the surface must be considered as a compressed ellipsoid of rotation, whose minor axis coincides with the earth's axis, and whose compression, flattening, or ellipticity is about 1/298.

_Measurement of Base Lines._

To determine by actual measurement on the ground the length of a side
of one of the triangles ("base line"), wherefrom to infer the lengths
of all the other sides in the triangulation, is not the least
difficult operation of a trigonometrical survey. When the problem is
stated thus--To determine the number of times that a certain standard
or unit of length is contained between two finely marked points on the
surface of the earth at a distance of some miles asunder, so that the
error of the result may be pronounced to lie between certain very
narrow limits,--then the question demands very serious consideration.
The representation of the unit of length by means of the distance
between two fine lines on the surface of a bar of metal at a certain
temperature is never itself free from uncertainty and probable error,
owing to the difficulty of knowing at any moment the precise
temperature of the bar; and the transference of this unit, or a
multiple of it, to a measuring bar will be affected not only with
errors of observation, but with errors arising from uncertainty of
temperature of both bars. If the measuring bar be not
self-compensating for temperature, its expansion must be determined by
very careful experiments. The thermometers required for this purpose
must be very carefully studied, and their errors of division and index
error determined.

In order to avoid the difficulty in exactly determining the
temperature of a bar by the mercury thermometer, F.W. Bessel
introduced in 1834 near Konigsberg a compound bar which constituted a
metallic thermometer.[1] A zinc bar is laid on an iron bar two toises
long, both bars being perfectly planed and in free contact, the zinc
bar being slightly shorter and the two bars rigidly united at one end.
As the temperature varies, the difference of the lengths of the bars,
as perceived by the other end, also varies, and affords a quantitative
correction for temperature variations, which is applied to reduce the
length to standard temperature. During the measurement of the base
line the bars were not allowed to come into contact, the interval
being measured by the insertion of glass wedges. The results of the
comparisons of four measuring rods with one another and with the
standards were elaborately computed by the method of least-squares.
The probable error of the measured length of 935 toises (about 6000
ft.) has been estimated as 1/863500 or 1.2 [mu] ([mu] denoting a
millionth). With this apparatus fourteen base lines were measured in
Prussia and some neighbouring states; in these cases a somewhat higher
degree of accuracy was obtained.

The principal triangulation of Great Britain and Ireland has seven
base lines: five have been measured by steel chains, and two, more
exactly, by the compensation bars of General T.F. Colby, an apparatus
introduced in 1827-1828 at Lough Foyle in Ireland. Ten base lines were
measured in India in 1831-1869 by the same apparatus. This is a system
of six compound-bars self-correcting for temperature. The bars may be
thus described: Two bars, one of brass and the other of iron, are laid
in parallelism side by side, firmly united at their centres, from
which they may freely expand or contract; at the standard temperature
they are of the same length. Let AB be one bar, A'B' the other; draw
lines through the corresponding extremities AA' (to P) and BB' (to Q),
and make A'P = B'Q, AA' being equal to BB'. If the ratio A'P/AP equals
the ratio of the coefficients of expansion of the bars A'B' and AB,
then, obviously, the distance PQ is constant (or nearly so). In the
actual instrument P and Q are finely engraved dots 10 ft. apart. In
practice the bars, when aligned, are not in contact, an interval of 6
in. being allowed between each bar and its neighbour. This distance is
accurately measured by an ingenious micrometrical arrangement
constructed on exactly the same principle as the bars themselves.

The last base line measured in India had a length of 8913 ft. In
consequence of some suspicion as to the accuracy of the compensation
apparatus, the measurement was repeated four times, the operations
being conducted so as to determine the actual values of the probable
errors of the apparatus. The direction of the line (which is at Cape
Comorin) is north and south. In two of the measurements the brass
component was to the west, in the others to the east; the differences
between the individual measurements and the mean of the four were
+0.0017, -0.0049, -0.0015, +0.0045 ft. These differences are very
small; an elaborate investigation of all sources of error shows that
the probable error of a base line in India is on the average [+-]2.8
[mu]. These compensation bars were also used by Sir Thomas Maclear in
the measurement of the base line in his extension of Lacaille's arc at
the Cape. The account of this operation will be found in a volume
entitled _Verification and Extension of Lacaille's Arc of Meridian at
the Cape of Good Hope_, by Sir Thomas Maclear, published in 1866. A
rediscussion has been given by Sir David Gill in his _Report on the
Geodetic Survey of South Africa, &c., 1896_.

A very simple base apparatus was employed by W. Struve in his
triangulations in Russia from 1817 to 1855. This consisted of four
wrought-iron bars, each two toises (rather more than 13 ft.) long; one
end of each bar is terminated in a small steel cylinder presenting a
slightly convex surface for contact, the other end carries a contact
lever rigidly connected with the bar. The shorter arm of the lever
terminates below in a polished hemisphere, the upper and longer arm
traversing a vertical divided arc. In measuring, the plane end of one
bar is brought into contact with the short arm of the contact lever
(pushed forward by a weak spring) of the next bar. Each bar has two
thermometers, and a level for determining the inclination of the bar
in measuring. The manner of transferring the end of a bar to the
ground is simply this: under the end of the bar a stake is driven very
firmly into the ground, carrying on its upper surface a disk, capable
of movement in the direction of the measured line by means of
slow-motion screws. A fine mark on this disk is brought vertically
under the end of the bar by means of a theodolite which is planted at
a distance of 25 ft. from the stake in a direction perpendicular to
the base. Struve investigated for each base the probable errors of the
measurement arising from each of these seven causes: Alignment,
inclination, comparisons with standards, readings of index, personal
errors, uncertainties of temperature, and the probable errors of
adopted rates of expansion. He found that [+-]0.8 [mu] was the mean of
the probable errors of the seven bases measured by him. The
Austro-Hungarian apparatus is similar; the distance of the rods is
measured by a slider, which rests on one of the ends of each rod.
Twenty-two base lines were measured in 1840-1899.

General Carlos Ibanez employed in 1858-1879, for the measurement of
nine base lines in Spain, two apparatus similar to the apparatus
previously employed by Porro in Italy; one is complicated, the other
simplified. The first, an apparatus of the brothers Brunner of Paris,
was a thermometric combination of two bars, one of platinum and one of
brass, in length 4 metres, furnished with three levels and four
thermometers. Suppose A, B, C three micrometer microscopes very firmly
supported at intervals of 4 metres with their axes vertical, and
aligned in the plane of the base line by means of a transit
instrument, their micrometer screws being in the line of measurement.
The measuring bar is brought under say A and B, and those micrometers
read; the bar is then shifted and brought under B and C. By repetition
of this process, the reading of a micrometer indicating the end of
each position of the bar, the measurement is made.

Quite similar apparatus (among others) has been employed by the French
and Germans. Since, however, it only permitted a distance of about 300
m. to be measured daily, Ibanez introduced a simplification; the
measuring rod being made simply of steel, and provided with inlaid
mercury thermometers. This apparatus was used in Switzerland for the
measurement of three base lines. The accuracy is shown by the
estimated probable errors: [+-]0.2 [mu] to [+-]0.8 [mu]. The distance
measured daily amounts at least to 800 m.

A greater daily distance can be measured with the same accuracy by
means of Bessel's apparatus; this permits the ready measurement of
2000 m. daily. For this, however, it is important to notice that a
large staff and favourable ground are necessary. An important
improvement was introduced by Edward Jaderin of Stockholm, who
measures with stretched wires of about 24 metres long; these wires are
about 1.65 mm. in diameter, and when in use are stretched by an
accurate spring balance with a tension of 10 kg.[2] The nature of the
ground has a very trifling effect on this method. The difficulty of
temperature determinations is removed by employing wires made of
invar, an alloy of steel (64%) and nickel (36%) which has practically
no linear expansion for small thermal changes at ordinary
temperatures; this alloy was discovered in 1896 by Benoit and
Guillaume of the International Bureau of Weights and Measures at
Breteuil. Apparently the future of base-line measurements rests with
the invar wires of the Jaderin apparatus; next comes Porro's apparatus
with invar bars 4 to 5 metres long.

Results have been obtained in the United States, of great importance
in view of their accuracy, rapidity of determination and economy. For
the measurement of the arc of meridian in longitude 98 deg. E., in
1900, nine base lines of a total length of 69.2 km. were measured in
six months. The total cost of one base was $1231. At the beginning and
at the end of the field-season a distance of exactly 100 m. was
measured with R.S. Woodward's "5-m. ice-bar" (invented in 1891); by
means of the remeasurement of this length the standardization of the
apparatus was done under the same conditions as existed in the case of
the base measurements. For the measurements there were employed two
steel tapes of 100 m. long, provided with supports at distances of 25
m., two of 50 m., and the duplex apparatus of Eimbeck, consisting of
four 5-m. rods. Each base was divided into sections of about 1000 m.;
one of these, the "test kilometre," was measured with all the five
apparatus, the others only with two apparatus, mostly tapes. The
probable error was about [+-]0.8 [mu], and the day's work a distance
of about 2000 m. Each of the four rods of the duplex apparatus
consists of two bars of brass and steel. Mercury thermometers are
inserted in both bars; these serve for the measurement of the length
of the base lines by each of the bars, as they are brought into their
consecutive positions, the contact being made by an elastic-sliding
contact. The length of the base lines may be calculated for each bar
only, and also by the supposition that both bars have the same
temperature. The apparatus thus affords three sets of results, which
mutually control themselves, and the contact adjustments permit rapid
work. The same device has been applied to the older
bimetallic-compensating apparatus of Bache-Wurdemann (six bases,
1847-1857) and of Schott. There was also employed a single rod
bimetallic apparatus on F. Porro's principle, constructed by the
brothers Repsold for some base lines. Excellent results have been more
recently obtained with invar tapes.

The following results show the lengths of the same German base lines
as measured by different apparatus:

metres.
Base at Berlin 1864 Apparatus of Bessel 2336.3920
" " 1880 " Brunner .3924
Base at Strehlen 1854 " Bessel 2762.5824
" " 1879 " Brunner .5852
Old base at Bonn 1847 " Bessel 2133.9095
" " 1892 " " .9097
New base at Bonn 1892 " " 2512.9612
" " 1892 " Brunner .9696

It is necessary that the altitude above the level of the sea of every
part of a base line be ascertained by spirit levelling, in order that
the measured length may be reduced to what it would have been had the
measurement been made on the surface of the sea, produced in
imagination. Thus if l be the length of a measuring bar, h its height
at any given position in the measurement, r the radius of the earth,
then the length radially projected on to the level of the sea is l(1 -
h/r). In the Salisbury Plain base line the reduction to the level of
the sea is -0.6294 ft.

The total number of base lines measured in Europe up to the present
time is about one hundred and ten, nineteen of which do not exceed in
length 2500 metres, or about 1-1/2 miles, and three--one in France, the
others in Bavaria--exceed 19,000 metres. The question has been
frequently discussed whether or not the advantage of a long base is
sufficiently great to warrant the expenditure of time that it
requires, or whether as much precision is not obtainable in the end by
careful triangulation from a short base. But the answer cannot be
given generally; it must depend on the circumstances of each
particular case. With Jaderin's apparatus, provided with invar wires,
bases of 20 to 30 km. long are obtained without difficulty.

In working away from a base line ab, stations c, d, e, f are carefully
selected so as to obtain from well-shaped triangles gradually
increasing sides. Before, however, finally leaving the base line, it
is usual to verify it by triangulation thus: during the measurement
two or more points, as p, q (fig. 1), are marked in the base in
positions such that the lengths of the different segments of the line
are known; then, taking suitable external stations, as h, k, the
angles of the triangles bhp, phq, hqk, kqa are measured. From these
angles can be computed the ratios of the segments, which must agree,
if all operations are correctly performed, with the ratios resulting
from the measures. Leaving the base line, the sides increase up to
10, 30 or 50 miles occasionally, but seldom reaching 100 miles. The
triangulation points may either be natural objects presenting
themselves in suitable positions, such as church towers; or they may
be objects specially constructed in stone or wood on mountain tops or
other prominent ground. In every case it is necessary that the precise
centre of the station be marked by some permanent mark. In India no
expense is spared in making permanent the principal trigonometrical
stations--costly towers in masonry being erected. It is essential that
every trigonometrical station shall present a fine object for
observation from surrounding stations.

_Horizontal Angles._

In placing the theodolite over a station to be observed from, the
first point to be attended to is that it shall rest upon a perfectly
solid foundation. The method of obtaining this desideratum must depend
entirely on the nature of the ground; the instrument must if possible
be supported on rock, or if that be impossible a solid foundation must
be obtained by digging. When the theodolite is required to be raised
above the surface of the ground in order to command particular points,
it is necessary to build two scaffolds,--the outer one to carry the
observatory, the inner one to carry the instrument,--and these two
edifices must have no point of contact. Many cases of high scaffolding
have occurred on the English Ordnance Survey, as for instance at
Thaxted church, where the tower, 80 ft. high, is surmounted by a spire
of 90 ft. The scaffold for the observatory was carried from the base
to the top of the spire; that for the instrument was raised from a
point of the spire 140 ft. above the ground, having its bearing upon
timbers passing through the spire at that height. Thus the instrument,
at a height of 178 ft. above the ground, was insulated, and not
affected by the action of the wind on the observatory.

At every station it is necessary to examine and correct the
adjustments of the theodolite, which are these: the line of
collimation of the telescope must be perpendicular to its axis of
rotation; this axis perpendicular to the vertical axis of the
instrument; and the latter perpendicular to the plane of the horizon.
The micrometer microscopes must also measure correct quantities on the
divided circle or circles. The method of observing is this. Let A, B,
C ... be the stations to be observed taken in order of azimuth; the
telescope is first directed to A and the cross-hairs of the telescope
made to bisect the object presented by A, then the microscopes or
verniers of the horizontal circle (also of the vertical circle if
necessary) are read and recorded. The telescope is then turned to B,
which is observed in the same manner; then C and the other stations.
Coming round by continuous motion to A, it is again observed, and the
agreement of this second reading with the first is some test of the
stability of the instrument. In taking this round of angles--or "arc,"
as it is called on the Ordnance Survey--it is desirable that the
interval of time between the first and second observations of A should
be as small as may be consistent with due care. Before taking the next
arc the horizontal circle is moved through 20 deg. or 30 deg.; thus a
different set of divisions of the circle is used in each arc, which
tends to eliminate the errors of division.

It is very desirable that all arcs at a station should contain one
point in common, to which all angular measurements are thus
referred,--the observations on each arc commencing and ending with
this point, which is on the Ordnance Survey called the "referring
object." It is usual for this purpose to select, from among the points
which have to be observed, that one which affords the best object for
precise observation. For mountain tops a "referring object" is
constructed of two rectangular plates of metal in the same vertical
plane, their edges parallel and placed at such a distance apart that
the light of the sky seen through appears as a vertical line about 10"
in width. The best distance for this object is from 1 to 2 miles.

This method seems at first sight very advantageous; but if, however,
it be desired to attain the highest accuracy, it is better, as shown
by General Schreiber of Berlin in 1878, to measure only single angles,
and as many of these as possible between the directions to be
determined. Division-errors are thus more perfectly eliminated, and
errors due to the variation in the stability, &c., of the instruments
are diminished. This method is rapidly gaining precedence.

The theodolites used in geodesy vary in pattern and in size--the
horizontal circles ranging from 10 in. to 36 in. in diameter. In
Ramsden's 36-in. theodolite the telescope has a focal length of 36 in.
and an aperture of 2.5 in., the ordinarily used magnifying power being
54; this last, however, can of course be changed at the requirements
of the observer or of the weather. The probable error of a single
observation of a fine object with this theodolite is about 0".2. Fig.
2 represents an altazimuth theodolite of an improved pattern used on
the Ordnance Survey. The horizontal circle of 14-in. diameter is read
by three micrometer microscopes; the vertical circle has a diameter of
12 in., and is read by two microscopes. In the great trigonometrical
survey of India the theodolites used in the more important parts of
the work have been of 2 and 3 ft. diameter--the circle read by five
equidistant microscopes. Every angle is measured twice in each
position of the zero of the horizontal circle, of which there are
generally ten; the entire number of measures of an angle is never
less than 20. An examination of 1407 angles showed that the probable
error of an observed angle is on the average [+-] 0".28.

For the observations of very distant stations it is usual to employ a
heliotrope (from the Gr. [Greek: helios], sun; [Greek: tropos], a
turn), invented by Gauss at Gottingen in 1821. In its simplest form
this is a plane mirror, 4, 6, or 8 in. in diameter, capable of
rotation round a horizontal and a vertical axis. This mirror is placed
at the station to be observed, and in fine weather it is kept so
directed that the rays of the sun reflected by it strike the distant
observing telescope. To the observer the heliotrope presents the
appearance of a star of the first or second magnitude, and is
generally a pleasant object for observing.

Observations at night, with the aid of light-signals, have been
repeatedly made, and with good results, particularly in France by
General Francois Perrier, and more recently in the United States by
the Coast and Geodetic Survey; the signal employed being an acetylene
bicycle-lamp, with a lens 5 in. in diameter. Particularly noteworthy
are the trigonometrical connexions of Spain and Algeria, which were
carried out in 1879 by Generals Ibanez and Perrier (over a distance of
270 km.), of Sicily and Malta in 1900, and of the islands of Elba and
Sardinia in 1902 by Dr Guarducci (over distances up to 230 km.); in
these cases artificial light was employed: in the first case electric
light and in the two others acetylene lamps.

_Astronomical Observations._

The direction of the meridian is determined either by a theodolite or
a portable transit instrument. In the former case the operation
consists in observing the angle between a terrestrial
object--generally a mark specially erected and capable of illumination
at night--and a close circumpolar star at its greatest eastern or
western azimuth, or, at any rate, when very near that position. If the
observation be made t minutes of time before or after the time of
greatest azimuth, the azimuth then will differ from its maximum value
by (450t)^2 sin 1" sin 2[delta]/ sin z, in seconds of angle, omitting
smaller terms, [delta] being the star's declination and z its zenith
distance. The collimation and level errors are very carefully
determined before and after these observations, and it is usual to
arrange the observations by the reversal of the telescope so that
collimation error shall disappear. If b, c be the level and
collimation errors, the correction to the circle reading is b cot z
[+-] c cosec z, b being positive when the west end of the axis is
high. It is clear that any uncertainty as to the real state of the
level will produce a corresponding uncertainty in the resulting value
of the azimuth,--an uncertainty which increases with the latitude and
is very large in high latitudes. This may be partly remedied by
observing in connexion with the star its reflection in mercury. In
determining the value of "one division" of a level tube, it is
necessary to bear in mind that in some the value varies considerably
with the temperature. By experiments on the level of Ramsden's 3-foot
theodolite, it was found that though at the ordinary temperature of 66
deg. the value of a division was about one second, yet at 32 deg. it
was about five seconds.

In a very excellent portable transit used on the Ordnance Survey, the
uprights carrying the telescope are constructed of mahogany, each
upright being built of several pieces glued and screwed together; the
base, which is a solid and heavy plate of iron, carries a reversing
apparatus for lifting the telescope out of its bearings, reversing it
and letting it down again. Thus is avoided the change of temperature
which the telescope would incur by being lifted by the hands of the
observer. Another form of transit is the German diagonal form, in
which the rays of light after passing through the object-glass are
turned by a total reflection prism through one of the transverse arms
of the telescope, at the extremity of which arm is the eye-piece. The
unused half of the ordinary telescope being cut away is replaced by a
counterpoise. In this instrument there is the advantage that the
observer without moving the position of his eye commands the whole
meridian, and that the level may remain on the pivots whatever be the
elevation of the telescope. But there is the disadvantage that the
flexure of the transverse axis causes a variable collimation error
depending on the zenith distance of the star to which it is directed;
and moreover it has been found that in some cases the personal error
of an observer is not the same in the two positions of the telescope.

To determine the direction of the meridian, it is well to erect two
marks at nearly equal angular distances on either side of the north
meridian line, so that the pole star crosses the vertical of each mark
a short time before and after attaining its greatest eastern and
western azimuths.

If now the instrument, perfectly levelled, is adjusted to have its
centre wire on one of the marks, then when elevated to the star, the
star will traverse the wire, and its exact position in the field at
any moment can be measured by the micrometer wire. Alternate
observations of the star and the terrestrial mark, combined with
careful level readings and reversals of the instrument, will enable
one, even with only one mark, to determine the direction of the
meridian in the course of an hour with a probable error of less than a
second. The second mark enables one to complete the station more
rapidly and gives a check upon the work. As an instance, at Findlay
Seat, in latitude 57 deg. 35', the resulting azimuths of the two marks
were 177 deg. 45' 37".29 [+-] 0".20 and 182 deg. 17' 15".61 [+-] 0".13,
while the angle between the two marks directly measured by a
theodolite was found to be 4 deg. 31' 37".43 [+-] 0".23.

We now come to the consideration of the determination of time with the
transit instrument. Let fig. 3 represent the sphere stereographically
projected on the plane of the horizon,--ns being the meridian, we the
prime vertical, Z, P the zenith and the pole. Let p be the point in
which the production of the axis of the instrument meets the celestial
sphere, S the position of a star when observed on a wire whose
distance from the collimation centre is c. Let a be the azimuthal
deviation, namely, the angle wZp, b the level error so that Zp = 90
deg. - b. Let also the hour angle corresponding to p be 90 deg. - n,
and the declination of the same = m, the star's declination being
[delta], and the latitude [phi]. Then to find the hour angle ZPS =
[tau] of the star when observed, in the triangles pPS, pPZ we have,
since pPS = 90 + [tau] - n,

-Sin c = sin m sin [delta] + cos m cos [delta] sin (n - [tau]),
Sin m = sin b sin [phi] - cos b cos [phi] sin a,
Cos m sin n = sin b cos [phi] + cos b sin [phi] sin a.

And these equations solve the problem, however large be the errors of
the instrument. Supposing, as usual, a, b, m, n to be small, we have
at once [tau] = n + c sec [delta] + m tan [delta], which is the
correction to the observed time of transit. Or, eliminating m and n by
means of the second and third equations, and putting z for the zenith
distance of the star, t for the observed time of transit, the
corrected time is t + (a sin z + b cos z + c) / cos [delta]. Another
very convenient form for stars near the zenith is [tau] = b sec [phi]
+ c sec [delta] + m (tan [delta] - tan [phi]).

Suppose that in commencing to observe at a station the error of the
chronometer is not known; then having secured for the instrument a
very solid foundation, removed as far as possible level and
collimation errors, and placed it by estimation nearly in the
meridian, let two stars differing considerably in declination be
observed--the instrument not being reversed between them. From these
two stars, neither of which should be a close circumpolar star, a good
approximation to the chronometer error can be obtained; thus let
[epsilon]1, [epsilon]2, be the apparent clock errors given by these
stars if [delta]1, [delta]2 be their declinations the real error is

[epsilon] = [epsilon]1 + ([epsilon]1 - [epsilon]2)
(tan [phi] - tan [delta]1) / (tan [delta]1 - tan [delta]2).

Of course this is still only approximate, but it will enable the
observer (who by the help of a table of natural tangents can compute
[epsilon] in a few minutes) to find the meridian by placing at the
proper time, which he now knows approximately, the centre wire of his
instrument on the first star that passes--not near the zenith.

The transit instrument is always reversed at least once in the course
of an evening's observing, the level being frequently read and
recorded. It is necessary in most instruments to add a correction for
the difference in size of the pivots.

The transit instrument is also used in the prime vertical for the
determination of latitudes. In the preceding figure let q be the point
in which the northern extremity of the axis of the instrument produced
meets the celestial sphere. Let nZq be the azimuthal deviation = a,
and b being the level error, Zq = 90 deg. - b; let also nPq = [tau]
and Pq = [psi]. Let S' be the position of a star when observed on a
wire whose distance from the collimation centre is c, positive when to
the south, and let h be the observed hour angle of the star, viz.
ZPS'. Then the triangles qPS', gPZ give

-Sin c = sin [delta] cos [psi] - cos [delta] sin [psi] cos (h + [tau]),
Cos [psi] = sin b sin [phi] + cos b cos [phi] cos a,
Sin [psi] sin [tau] = cos b sin a.

Now when a and b are very small, we see from the last two equations
that [psi] = [phi] - b, a = [tau] sin [psi], and if we calculate
[phi]' by the formula cot [phi]' = cot [delta] cos h, the first
equation leads us to this result--

[phi] = [phi]' + (a sin z + b cos z + c)/cos z,

the correction for instrumental error being very similar to that
applied to the observed time of transit in the case of meridian
observations. When a is not very small and z is small, the formulae
required are more complicated.

The method of determining latitude by transits in the prime vertical
has the disadvantage of being a somewhat slow process, and of
requiring a very precise knowledge of the time, a disadvantage from
which the zenith telescope is free. In principle this instrument is
based on the proposition that when the meridian zenith distances of
two stars at their upper culminations--one being to the north and the
other to the south of the zenith--are equal, the latitude is the mean
of their declinations; or, if the zenith distance of a star
culminating to the south of the zenith be Z, its declination being
[delta], and that of another culminating to the north with zenith
distance Z' and declination [delta]', then clearly the latitude is
1/2([delta] + [delta]') + 1/2(Z - Z'). Now the zenith telescope does
away with the divided circle, and substitutes the measurement
micrometrically of the quantity Z' - Z.

In fig. 4 is shown a zenith telescope by H. Wanschaff of Berlin, which
is the type used (according to the Central Bureau at Potsdam) since
about 1890 for the determination of the variations of latitude due to
different, but as yet imperfectly understood, influences. The
instrument is supported on a strong tripod, fitted with levelling
screws; to this tripod is fixed the azimuth circle and a long vertical
steel axis. Fitting on this axis is a hollow axis which carries on its
upper end a short transverse horizontal axis with a level. This latter
carries the telescope, which, supported at the centre of its length,
is free to rotate in a vertical plane. The telescope is thus mounted
eccentrically with respect to the vertical axis around which it
revolves. Two extremely sensitive levels are attached to the
telescope, which latter carries a micrometer in its eye-piece, with a
screw of long range for measuring differences of zenith distance. Two
levels are employed for controlling and increasing the accuracy. For
this instrument stars are selected in pairs, passing north and south
of the zenith, culminating within a few minutes of time and within
about twenty minutes (angular) of zenith distance of each other. When
a pair of stars is to be observed, the telescope is set to the mean of
the zenith distances and in the plane of the meridian. The first star
on passing the central meridional wire is bisected by the micrometer;
then the telescope is rotated very carefully through 180 deg. round
the vertical axis, and the second star on passing through the field is
bisected by the micrometer on the centre wire. The micrometer has thus
measured the difference of the zenith distances, and the calculation
to get the latitude is most simple. Of course it is necessary to read
the level, and the observations are not necessarily confined to the
centre wire. In fact if n, s be the north and south readings of the
level for the south star, n', s' the same for the north star, l the
value of one division of the level, m the value of one division of the
micrometer, r, r' the refraction corrections, [mu], [mu]' the
micrometer readings of the south and north star, the micrometer being
supposed to read from the zenith, then, supposing the observation made
on the centre wire,--

[phi] = 1/2([delta] + [delta]') + 1/2([mu] - [mu]')m + 1/4(n + n' - s - s')l +
1/2(r - r').

It is of course of the highest importance that the value m of the
screw be well determined. This is done most effectually by observing
the vertical movement of a close circumpolar star when at its greatest
azimuth.

In a single night with this instrument a very accurate result, say
with a probable error of about 0".2, could be obtained for latitude
from, say, twenty pair of stars; but when the latitude is required to
be obtained with the highest possible precision, two nights at least
are necessary. The weak point of the zenith telescope lies in the
circumstance that its requirements prevent the selection of stars
whose positions are well fixed; very frequently it is necessary to
have the declinations of the stars selected for this instrument
specially observed at fixed observatories. The zenith telescope is
made in various sizes from 30 to 54 in. in focal length; a 30-in.
telescope is sufficient for the highest purposes and is very portable.
The net observation probable-error for one pair of stars is only
[+-]0".1.

The zenith telescope is a particularly pleasant instrument to work
with, and an observer has been known (a sergeant of Royal Engineers,
on one occasion) to take every star in his list during eleven hours on
a stretch, namely, from 6 o'clock P.M. until 5 A.M., and this on a
very cold November night on one of the highest points of the
Grampians. Observers accustomed to geodetic operations attain
considerable powers of endurance. Shortly after the commencement of
the observations on one of the hills in the Isle of Skye a storm
carried away the wooden houses of the men and left the observatory
roofless. Three observatory roofs were subsequently demolished, and
for some time the observatory was used without a roof, being filled
with snow every night and emptied every morning. Quite different,
however, was the experience of the same party when on the top of Ben
Nevis, 4406 ft. high. For about a fortnight the state of the
atmosphere was unusually calm, so much so, that a lighted candle could
often be carried between the tents of the men and the observatory,
whilst at the foot of the hill the weather was wild and stormy.

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 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 an electric 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 chronographs. 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.

These errors can nevertheless be almost entirely avoided by using the
impersonal micrometer of Dr Repsold (Hamburg, 1889). In this device
there is a movable micrometer wire which is brought by hand into
coincidence with the star and moved along with it; at fixed points
there are electrical contacts, which replace the fixed wires.
Experiments at the Geodetic Institute and Central Bureau at Potsdam in
1891 gave the following personal equations in the case of four
observers:--

Older Procedure. New Procedure.

A-B -0^s.108 -0^s.004
A-G -0^s.314 -0^s.035
A-S -0^s.184 -0^s.027
B-G -0^s.225 +0^s.013
B-S -0^s.086 -0^s.023
G-S +0^s.109 -0^s.006

These results show that in the later method the personal equation is
small and not so variable; and consequently the repetition of
longitude determinations with exchanged observers and apparatus
entirely eliminates the constant errors, the probable error of such
determinations on ten nights being scarcely [+-]0^s.01.

_Calculation of Triangulation._

The surface of Great Britain and Ireland is uniformly covered by
triangulation, of which the sides are of various lengths from 10 to
111 miles. The largest triangle has one angle at Snowdon in Wales,
another on Slieve Donard in Ireland, and a third at Scaw Fell in
Cumberland; each side is over a hundred miles and the spherical excess
is 64". The more ordinary method of triangulation is, however, that of
chains of triangles, in the direction of the meridian and
perpendicular thereto. The principal triangulations of France, Spain,
Austria and India are so arranged. Oblique chains of triangles are
formed in Italy, Sweden and Norway, also in Germany and Russia, and in
the United States. Chains are composed sometimes merely of consecutive
plain triangles; sometimes, and more frequently in India, of
combinations of triangles forming consecutive polygonal figures. In
this method of triangulating, the sides of the triangles are generally
from 20 to 30 miles in length--seldom exceeding 40.

The inevitable errors of observation, which are inseparable from all
angular as well as other measurements, introduce a great difficulty
into the calculation of the sides of a triangulation. Starting from a
given base in order to get a required distance, it may generally be
obtained in several different ways--that is, by using different sets
of triangles. The results will certainly differ one from another, and
probably no two will agree. The experience of the computer will then
come to his aid, and enable him to say which is the most trustworthy
result; but no experience or ability will carry him through a large
network of triangles with anything like assurance. The only way to
obtain trustworthy results is to employ the method of least squares.
We cannot here give any illustration of this method as applied to
general triangulation, for it is most laborious, even for the simplest
cases.

Three stations, projected on the surface of the sea, give a spherical
or spheroidal triangle according to the adoption of the sphere or the
ellipsoid as the form of the surface. 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
[alpha], [beta], [gamma] respectively. Now the angle A of the triangle
as measured by a theodolite is the inclination of the planes BA[alpha]
and CA[alpha], and the angle at B is that contained by the planes
AB[beta] and CB[beta]. But the planes AB[alpha] and AB[beta]
containing the line AB in common cut the surface in two distinct plane
curves. In order, therefore, that 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 geodetic or shortest lines.
C.C.G. Andrae, of Copenhagen, has also shown that other lines give a
less convenient computation.

K.F. Gauss, in his treatise, _Disquisitiones generales circa
superficies curvas_, entered fully into the subject of geodetic (or
geodesic) triangles, and investigated expressions for the angles of a
geodetic 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 geodetic triangle, while A, B,
C are those of a plane triangle having sides equal respectively to
those of the geodetic triangle, then, [sigma] being the area of the
plane triangle and [a], [b], [c] the measures of curvature at the
angular points,

[A] = A + [sigma](2[a] + [b] + [c])/12,
[B] = B + [sigma]([a] + 2[b] + [c])/12,
[C] = C + [sigma]([a] + [b] + 2[c])/12.

For the sphere [a] = [b] = [r], and making this simplification, we
obtain the theorem previously given by A.M. Legendre. With the terms
of the fourth order, we have (after Andrae):

[epsilon] [sigma] /m^2 - a^2 [a] - k \
[A] - A = --------- + -------k ( ---------k + ------- ),
3 3 \ 20 4k /

[epsilon] [sigma] /m^2 - b^2 [b] - k \
[B] - B = --------- + -------k ( ---------k + -------- ),
3 3 \ 20 4k /

[epsilon] [sigma] /m^2 - c^2 [c] - k \
[C] - C = --------- + -------k ( ---------k + -------- ),
3 3 \ 20 4k /

in which [epsilon] = [sigma] k {1 + (m^2k / 8)}, 3m^2 = a^2 + b^2 +
c^2, 3k = [a] + [b] + [c]. For the ellipsoid of rotation the measure
of curvature is equal to 1 / [rho]n, [rho] and n being the radii of
curvature of the meridian and perpendicular.

It is rarely that the terms of the fourth order are required. As a
rule spheroidal triangles are calculated as spherical (after
Legendre), i.e. like plane triangles with a decrease of each angle of
about [epsilon] / 3; [epsilon] must, however, be calculated for each
triangle separately with its mean measure of curvature k.

The geodetic 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 a plane drawn perpendicular to pq, and let S be any point in
the intersection of this plane with the surface; then 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 geodetic 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
co-ordinates of B, those of A and C will be respectively--

x - cl : y - cm : z - cn
x + cl': y + cm': z + cn'.

Hence the co-ordinates of the middle point M of AC are x + 1/2c(l' - l),
y + 1/2c(m' - m), z + 1/2c(n' - n), and the direction cosines of BM are
therefore proportional to l' - l : m' - m : n' - n. If the angle made
by BC with AB be indefinitely small, the direction cosines of BM are
as [delta]l : [delta]m : [delta]n. Now if AB, BC be two contiguous
elements of a geodetic, then BM must be a normal to the surface, and
since [delta]l, [delta]m, [delta]n are in this case represented by
[delta](dx/ds), [delta](dy/ds), [delta](dz/ds), and if the equation of
the surface be u = 0, we have

d^2x / du d^2y / du d^2z / du
---- / -- = ---- / -- = ---- / --,
ds^2 / dx ds^2 / dy ds^2 / dz

which, however, are equivalent to only one equation. In the case of
the spheroid this equation becomes

d^2x d^2y
y ---- - x ---- = 0,
ds^2 ds^2

which integrated gives ydx - xdy = Cds. This again may be put in the
form r sin a = C, where a is the azimuth of the geodetic at any
point--the angle between its direction and that of the meridian--and r
the distance of the point from the axis of revolution.