The geography and geology of south-eastern Egypt

Chapter III (1)

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=SURVEYING METHODS AND PRINCIPAL GEOGRAPHICAL RESULTS.=
* * * * *

=Triangulation.=

=Base-Lines=—The triangulation was commenced by measuring a base line[62] near Gebel Muelih by means of a 100-metre steel tape, which had been previously standardised at the Khedivial Observatory. A level tract having been selected, in such a position as to afford easy connection with points already triangulated from the Nile Valley, a line about two and a half kilometres long was ranged out, along which wooden pegs were driven flush with the ground at 100-metre intervals. The inequalities of the ground between the pegs were levelled off by tiny embankments and cuttings, so as to enable the tape to lie flat. On each peg was nailed a zinc plate having a millimetre scale running in the direction of the line. In measuring, the tape was laid on the ground, stretched to a constant tension by a spring balance at each end, in such manner that its end marks fell on the zinc scales, and readings were simultaneously taken on the scales at the two ends. Temperatures were taken in several places along the tape by mercurial thermometers. The operation of measurement was carried out in the early morning, so as to avoid any large difference of temperature between the ground and the air. The levels of the pegs were found by spirit levelling. After correction for the initial error of the tape, temperature, stretch, and inclination, the true length of the base-line reduced to sea-level was found, as the mean of two separate measurements at different tensions, to be 2,482·280 metres. The azimuth of the line was found by observations of _Polaris_ at elongation to be 30° 30′ 6″ E. of N. The geographical position of the west end of the line was found by triangulation-connexion with the Nile Valley by Mr. Villiers Stuart to be latitude 24° 53′ 36″·7 N., longitude 34° 4′ 17″·9 E.

A second connexion to a base-line was made near Gebel Um Harba, where a base had been previously measured by Mr. Villiers Stuart in the course of his triangulation of the western part of the desert. In this case the connexion was not made to the actual base-line, but to another main line tied directly on to it. The data at this point of connexion from the two triangulations afforded a useful check on the accuracy of the work, and were as follows:—

=Reconnaissance for triangulation points= was carried out simultaneously with the triangulation itself. The distance of likely looking peaks was determined either by intersecting them from distant stations, or by special small triangulations from short bases, and such selection made as seemed most likely to secure well shaped figures and a good command of surrounding country. As a rule, the highest summits were selected as main (occupied) stations, while all other prominent peaks and other features were fixed by intersection from two or more main stations. The form adopted for the main triangulation net was a series of quadrilateral figures with diagonals, combined with centric polygons, all the angles of the figures being generally measured. The average length of side was about thirty-five kilometres.

=Beacons.=—Main stations were marked by wrought-iron beacons, consisting of two lengths of stove piping about 15 centimetres diameter by 1½ metres long, the upper length fitting into a faucet made by splaying out the lower tube. Near the top of the tube were affixed four sheet iron wings, bolted on to angle iron cleats. A conical cairn of stones was built up round the tube, nearly up to the wings, so that the beacon when erected was about three metres high, two metres in diameter across the base of the cairn, and about a metre wide across the wings. The beacons were taken down while a station was being occupied, and replaced on leaving.

Intersected points were sometimes marked with a beacon or cairn, but in general the peaks were simply bisected from several stations, as it was found that this gave sufficiently accurate results.

=Measurement of Horizontal Angles.=—Angles were measured with a 6-inch theodolite furnished with reading microscopes graduated to 10″ and permitting of reliable estimations to 1″. The angles between main points were read on four arcs to eliminate circle errors. Intersected points were observed on one arc only. The average error of closure of main triangles was 3″·3.

=Field Computation and Plotting of Triangulated Points.=—The triangles were computed by the ordinary method, but the angles were rounded off to 10″ to enable the sines to be taken direct from the logarithmic tables, and the logarithms were only taken to five places. The length of the sides having been thus found, the geographical positions were found by the ordinary _L M Z_ computation, using, however, only two latitude terms and 5-place logarithms, while azimuths were only taken out to the nearest 10″. The abbreviated form of computation used will be best illustrated by an example:—[63]

_Computation of Position of_ △ _No. 260 from No. 275._

_l_ = 20042 m. { φ = 23° 55′ 30″·6
275 {
_Z_ = 64° 29′ 20″ E of N. { λ = 34° 54′ 36″·9

Log _l_ = 4·30194 Log _l_² = 8·604
- -
Log cos _Z_ = 1·63416 Log sin² _Z_ = 1·911
- -
_B_ 2·51194 _C_ = 9·053
------- -----
-
2·44804 = 1·568
------- -----
_d_φ, 1st term = + 280″·6 Log _l_ = 4·30194
-
2nd term = − 0″·4 Log sin _Z_ = 1·95545
-------------- -
_d_φ = 280″·2 _A′_ = 2·50948
-------
= 4′ 40″·2 = 2·76687
-
φ = 23° 55′ 30″·6 Log cos φ′ = 1·96072
-------------- -------
φ′ = 24° 0′ 10″·8 = 2·80615
-------------- -------
_d_λ = 640″·0

Whence = 10′ 40″·0

{ φ′ = 24° 0′ 10″·8 λ = 34° 54′ 36″·9
260 { --------------
{ λ′ = 35° 5′ 16″·9 λ′ = 35° 5′ 16″·9
--------------

In the above, it will be noticed that the azimuth is always noted as so much east or west of north or south. If this convention be adopted, one may consider the first term of _d_φ as always +, and the second term will be + if the azimuth contains the word south,—if it is from the north, while the total _d_φ is to be added or subtracted according as one is going north or south. A somewhat similar convention is adopted in neglecting the sign of _d_λ till the actual addition or subtraction is made. It was found in the field that this method prevented any mistake of sign, while being much simpler to work than one involving angles greater than 90°.

The geographical coordinates thus found were plotted directly on to the plane-table sheets, on which the graticule at 10′ intervals was the first thing drawn. The odd minutes and seconds were first converted into minutes and decimals, and then into kilometres by multiplying by the factors appropriate to the latitude, so that the plotting could be done by the ordinary scale of kilometres. To avoid difficulties of paper-shrinkage, as many points as possible were plotted at the time of drawing the graticule, and in general the points had to be plotted as far ahead as possible for controlling the traversing and sketching.

=Astronomical Observations.=—Astronomical checks on the triangulation were obtained by observations of latitude at certain selected main stations 60-120 kilometres apart, and by azimuth observations for certain main lines.

The method used for =latitude= was that of observing the times of equal altitudes of three or more stars, selected as near to the meridian[64] as possible. This method presents great advantages over the usual _Polaris_ and circummeridian altitudes, in that the observations are more easily made, and yield much more accurate results, because uncertainties in refraction are largely eliminated and the errors of circle graduation are not involved, the altitudes not being read at all.[65] The theodolite used was the same as was employed in triangulation, and the times were taken by a half chronometer watch, preferably one marking sidereal time with a rate which could be considered negligible during the hour or so occupied by the observation. The first star taken was usually _Polaris_, and the vertical circle was left clamped at its altitude. For the other stars, any dislevelment was corrected by touching up the levelling screws just before the instant of observation; this was found better, than taking bubble readings and correcting for slight difference of altitude.

The method which I found best in the field for reducing the observations differs somewhat from that described by Chauvenet. Assuming approximate values for the latitude and watch error, I first calculated the altitude of each star from the formula

sin _h_ = sin φ sin ε + cos φ cos δ cos _t_

If the assumed latitude and watch error were correct, all the stars would give the same value for _h_. If not, each star would give an equation of the form

_h_ + cos _A_ _d_φ + cos φ sin _A_ _dT_ − _h_0 = 0

where _A_ is the star’s azimuth, _d_φ the required correction to the assumed latitude, _dT_ the required correction to the assumed watch times, and _h_0 the true altitude common to the three stars. The values of cos _A_ and cos φ sin _A_ were calculated from the ordinary formula

sin _A_ = (cos δ sin _t_)/cos _h_

by four-place logarithms (using the approximate values for φ, _t_, and _h_, since these are quite sufficiently accurate for the purpose) and inserted into the three star-equations.[66] By then solving the three simultaneous equations for _d_φ, the required correction to the assumed latitude was at once obtained.

As the method is one not usually treated of in books on practical astronomy, I give on the following pages the reduction of an observation worked out in full.

_Latitude by Equal Altitudes of Three Stars._

Station on Gebel Um Heshenib. January 30, 1906.

Approximate φ = 24° 20′ 50″ N.

„ λ = 34° 51′ 0″ E.

Sidereal watch U. and C. 30811, approximately 2m 22s fast on L.S.T., rate negligible.

Observed times of equal altitudes by watch:—

_Polaris_ 3h 33m 54s·2

α _Columbæ_ 3 42 40 ·8

ε _Canis majoris_ 4 24 35 ·8

_Polaris._

Watch time 3h 33m 54s·2

Watch fast 2 22 ·0
------------
L.S.T. 3 31 32 ·2

Star’s R.A. 1 25 9 ·1
------------
_t_ 2 6 23 ·1 = 31° 35′ 46″·5 W. of meridian.
------------

-
φ = 24° 20′ 50″ log sin 1·6151769
-
δ = 88° 48′ 33″·1 log sin 1·9999062
---------
- -
1·6150831 log cos δ 2·3177
-
Nat. (1) 0·4121764 log sin _t_ 1·7193
--------- ------
-
2·0370
- -
φ = 24° 20′ 50″ log cos 1·9595488 log cos _h_ 1·9560
------
- -
δ = 88° 48′ 33″·1 log cos 2·3176870 log sin _A_ 2·0810
- -
_t_ = 31° 35′ 46″·5 log cos 1·9303179 log cos φ 1·9595
--------- ------
-
2·0405
- -
2·2075537 Nat. = 0·011
---------
Nat. (2) 0·0161270

Nat. (1) 0·4121764 log cos _A_ 1·9999
---------
0·4283034 Nat. = 1·000
-
log sin _h_ 1·6317515

_h_ = 25° 21′ 35″·8

Whence the equation for _Polaris_ is

35·8 + 1·000 _d_φ − 0·011 _dT_ − _h_0 = 0 (1)

α _Columbæ._

Watch time 3h 42m 40s·8

Watch fast 2 22 ·0
------------
L.S.T. 3 40 18 ·8

Star’s R.A. 5 36 15 ·6
------------
_t_ 1 55 56 ·8 = 28° 59′ 12″·0 E. of meridian.
------------

-
φ = 24° 20′ 50″ log sin 1·6151769
-
δ = 34° 7′ 45″·7 log sin 1·7490118
---------
-
1·3641887
-
Nat. (1) 0·2313070 log cos δ 1·9179
--------- -
log sin _t_ 1·6854
------
- -
φ = 24° 20′ 50″ log cos 1·9595488 1·6033
- -
δ = 34° 7′ 45″·7 log cos 1·9179112 log cos _h_ 1·9560
------
- -
_t_ = 28° 59′ 12″·0 log cos 1·9418753 log sin _A_ 1·6473
---------
- -
1·8193353 log cos φ 1·9595
--------- ------
Nat. (2) 0·6596830 1·6068

Nat. (1) 0·2313070 Nat. = 0·404
---------
0·4283760
- -
log sin _h_ 1·6318196 log cos _A_ 1·9523

Nat. = 0·896

_h_ = 25° 21′ 51″·1

Whence the equation for α _Columbæ_ is

51·1 − 0·896 _d_φ + 0·404 _dT_ − _h_0 = 0 (2)

ε _Canis majoris._

Watch time 4h 24m 35s·8

Watch fast 2 22 ·0
------------
4 22 13 ·8

Star’s R.A. 6 54 57 ·1
------------
_t_ 2 32 43 ·3 = 38° 10′ 49″·5 E. of meridian.
------------

-
φ = 24° 20′ 50″ log sin 1·6151769
-
δ = 28° 50′ 52″·7 log sin 1·6834857
---------
-
1·2986626

Nat. (1) 0·1989128
---------
-
φ = 24° 20′ 50″ log cos 1·9595488
- -
δ = 28° 50′ 52″·7 log cos 1·9424561 log cos δ 1·9424

- -
_t_ = 38° 10′ 49″·5 log cos 1·8954602 log sin _t_ 1·7911
--------- ------
- -
1·7974651 1·7335
--------- -
Nat. (2) 0·6272853 log cos _h_ 1·9560
------
-
Nat. (1) 0·1989128 log sin _A_ 1·7775
--------- -
0·4283725 log cos φ 1·9595
- ------
log sin _h_ 1·6318215 1·7370

Nat. = 0·546

_h_ = 25° 21′ 51″·5
-
log cos _A_ 1·9034

Nat. = 0·801

Whence the equation for ε _Canis majoris_ is

51·5 − 0·801 _d_φ + 0·546 _dT_ − _h_0 = 0 (3)

Collecting the equations of the three stars, we have

35·8 + 1·000 _d_φ − 0·011 _dT_ − _h_0 = 0 (1)

51·1 − 0·896 _d_φ + 0·404 _dT_ − _h_0 = 0 (2)

51·5 − 0·801 _d_φ + 0·546 _dT_ − _h_0 = 0 (3)

By solving these equations for _d_φ, we find

_d_φ = + 6″·5

whence the latitude is found to be

24° 20′ 56″·5

It may be remarked that the above process was considerably shortened when it was possible to get a pair of observations on the same south star both east and west of the meridian, instead of on two separate south stars. In that case the watch error was found at once from the difference between the star’s R.A. and the mean of the two observed times, and the latitude could be found from two equations instead of three. The condition for this modification of the method was that a nautical almanac star could be found culminating at an altitude slightly greater than that of _Polaris_ at a time convenient for the observation. It is not advisable to select a star which would give too long an interval between the equal east and west altitudes. The best results are obtainable when the interval between the two observations of the south star is about an hour, and when _Polaris_ is near its transit. Under such circumstances the watch correction is obtained quite nearly enough for a good latitude; for, as Gauss[67] pointed out, “the essential condition is not so much that the precise instant when the star reaches a supposed place should be noted, as that at the time which is noted the star should not be sensibly distant from that place.”

The following table shows the latitudes found by observation and triangulation at the various points. It may be remarked that the method used for latitude determination was liable to observational errors of 2″ or so, as well as to errors of possibly more than double that amount due to plumb-line deflection among the mountains,[68] so that the observed latitudes were only taken as checks to prevent any gross error in triangulation being overlooked, and not for any determination of the figure of the earth, for which latter purpose more elaborate observations would have been necessary.

--------------------+--------------+--------------+-------------------
|Lat. observed.|Lat. computed | Difference
Point. | | from |computed-observed.
| |Triangulation.|
--------------------+--------------+--------------+-------------------
West Peg, Muelih |24° 53′ 40″·3 |24° 53′ 36″·7 | − 3″·6
Base | | |
| | |
Beacon on Gebel Um |24° 20′ 56″·5 |24° 20′ 49″·2 | − 7″·3
Heshenib | | |
| | |
„ „ Hill near|23° 55′ 33″·2 |23° 55′ 30″·6 | − 2″·6
Gebel Selaia | | |
| | |
„ „ Berenice |23° 54′ 39″·5 |23° 54′ 40″·3 | − 0″·8
Temple | | |
| | |
„ „ Gimeida |22° 46′ 33″·2 |22° 46′ 29″·4 | − 3″·8
Hill | | |
--------------------+--------------+--------------+-------------------

=Azimuths= were determined in the usual manner[69] by elongations of close circumpolar stars, _Polaris_ or _51 Cephei_ being usually selected. The azimuth mark used was an ordinary Egyptian _shamadan_ (candlestick with spring feed) with a glass globe, placed at a distance of one to two kilometres, with its foot firmly bedded in sand and stones to prevent any motion. The azimuths observed at the different stations are shown in the following table:—

The observed azimuths, unlike the observed latitudes, were more accurate than the results of triangulation, repetition having shown them to be reliable within 2″ or 3″, an error of which magnitude would soon be surpassed in the process of continuing a chain of azimuths with the unadjusted values of the angles of the triangles, which were the only values possible to be used in the field. On arriving at an azimuth station, therefore, a fresh chain of azimuths was begun from the results of the observation, and continued to the next station where astronomical observations were undertaken. The accumulated azimuth error was, however, never found to exceed 10″ in any chain, a quantity which could not sensibly affect the computed positions of points for plotting on the maps.

=Connexion with the Sudan Surveys.=—At the south end of the area, connexion was made to a number of points triangulated by the Sudan Surveys, but as the Sudan triangulation was commenced as an independent piece of work from an observed latitude and a telegraphically determined longitude, the connexion affords no check on the accuracy of the triangulations. The difference found between my positions and those of the Sudan Surveys was practically constant for all the Sudan points connected, and amounted to 3″·5 in latitude and 26″ in longitude[70]; these figures represent the errors in the assumed latitude and longitude of the starting point of the Sudan Surveys, and will be employed as corrections to the Sudan positions now that a complete chain of triangulation connects Berber with the Mediterranean.

=Levels of Triangulation Points.=—The altitudes above sea-level of all triangulation points were determined by vertical angular measurements carried out at the occupied stations, an actual sea-level datum being obtained by including rocks awash in the sea among the triangulated points. To secure constancy of atmospheric refraction as far as possible, vertical angles were always read in the middle of the day, where the change of refraction is slowest. For the occupied stations, refraction and curvature were eliminated by reciprocal observations. For intersected points the formula _h_ = _d_ _tan_ θ + (1 − _k_)/2_r_ _d_² was used, the value of _d_ tan θ being first found by five-figure logarithms and then that of the curvature and refraction correction (1 − _k_)/2_r_ _d_² by means of the very convenient “Universal” slide rule of Nessler.[71] The value of _k_ found from a discussion of the first few reciprocal observations was found to be very nearly 0·13, corresponding with the mean of European determinations, and this value for the coefficient was employed throughout the work for intersected points.[72] For obtaining the correction (1 − _k_)/2_r_ _d_² by the slide rule, a mark _R_ was scratched at 1210[73] on the lower scale of the slide; by bringing this mark _R_ opposite to the distance (in kilometres) on the lower scale (or, where the logarithm of the distance was more convenient, by bringing the mark vertically under that logarithm on the log scale of the rule by means of the cursor) the correction could be read off directly on the lowest fixed scale opposite the end-graduation of the slide. Usually four or five values for the altitude of a single point were obtained from a corresponding number of stations, and the mean taken; the various values generally agreed within two metres.

The constant combination of vertical angular measurements with horizontal ones was of great service from another point of view from that of providing altitude data for the maps. It frequently happened that a peak observed at one station could not be identified among a number of similar peaks visible at another station. When this trouble arose, the vertical angles offered a way out of the difficulty. Vertical and horizontal angles were read off to a number of likely-looking peaks; on working out the triangles to the nearest minute of the observed angles, the distance of each peak was obtained on the assumption that it was the one required. Then to find which of the several peaks was the correct one, the elevations were worked out, assuming the distances correct; in only one case would the level agree from the two stations, and this obviously discriminated the peak required. The working out of the triangles for this purpose could be done with sufficient accuracy in a very few minutes by means of the slide rule, and many points were thus saved from rejection consequent on misidentification.

Checks on absolute level were frequently obtained by observing depression angles to the sea horizon, using the formula θ = 107·8 √{_h_}, where θ is in seconds of arc and _h_ is the altitude in metres. But for high stations the horizon is so distant that very small variations in refraction cause rather large errors in the result, so that this method only furnished a rough check.

=Summary of Triangulated Positions.=

The following tables give the geographical positions and altitudes above sea-level of all points triangulated, arranged in order of diminishing latitude, _i.e._, from north to south. The list includes the points fixed within the Sudan for connexion with the Sudan Surveys. Stations occupied are indicated by an asterisk against the number of the point. It will be noticed that in some cases several different mountains bear the same name though widely distant from each other; also that where a single mountain mass possesses several peaks or summits, each of these has been fixed separately.

SUMMARY OF TRIANGULATED POSITIONS.

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