35·8 + 1·000dφ− 0·011dT−h0= 0(1)αColumbæ.Watch time3h42m40s·8Watch fast222·0L.S.T.34018·8Star’s R.A.53615·6t15556·8= 28° 59′ 12″·0 E. of meridian.φ = 24° 20′ 50″log sin1·6151769δ = 34°7′ 45″·7log sin1·74901181·3641887Nat. (1)0·2313070log cos δ1·9179log sint1·6854φ = 24° 20′ 50″log cos1·95954881·6033δ = 34°7′ 45″·7log cos1·9179112log cosh1·9560t= 28° 59′ 12″·0log cos1·9418753log sinA1·64731·8193353log cos φ1·9595Nat. (2)0·65968301·6068Nat. (1)0·2313070Nat. =0·4040·4283760log sinh1·6318196log cosA1·9523Nat. =0·896h=25° 21′ 51″·1Whence the equation for αColumbæis51·1 − 0·896dφ+ 0·404dT−h0= 0(2)εCanis majoris.Watch time4h24m35s·8Watch fast222·042213·8Star’s R.A.65457·1t23243·3= 38° 10′ 49″·5 E. of meridian.φ = 24° 20′ 50″log sin1·6151769δ = 28° 50′ 52″·7log sin1·68348571·2986626Nat. (1)0·1989128φ = 24° 20′ 50″log cos1·9595488δ = 28° 50′ 52″·7log cos1·9424561log cos δ1·9424t= 38° 10′ 49″·5log cos1·8954602log sint1·79111·79746511·7335Nat. (2)0·6272853log cosh1·9560Nat. (1)0·1989128log sinA1·77750·4283725log cos φ1·9595log sinh1·63182151·7370Nat. =0·546h=25° 21′ 51″·5log cosA1·9034Nat. =0·801Whence the equation for εCanis majorisis51·5 − 0·801dφ+ 0·546dT−h0= 0(3)Collecting the equations of the three stars, we have35·8 + 1·000dφ− 0·011dT−h0= 0(1)51·1 − 0·896dφ+ 0·404dT−h0= 0(2)51·5 − 0·801dφ+ 0·546dT−h0= 0(3)By solving these equations fordφ, we finddφ= + 6″·5whence the latitude is found to be24° 20′ 56″·5It 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 ofPolarisat 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 whenPolarisis 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.Point.Lat. observed.Lat. computed from Triangulation.Difference computed-observed.West Peg, Muelih Base24° 53′ 40″·324° 53′ 36″·7− 3″·6Beacon on Gebel Um Heshenib24° 20′ 56″·524° 20′ 49″·2− 7″·3„„Hill near Gebel Selaia23° 55′ 33″·223° 55′ 30″·6− 2″·6„„Berenice Temple23° 54′ 39″·523° 54′ 40″·3− 0″·8„„Gimeida Hill22° 46′ 33″·222° 46′ 29″·4− 3″·8Azimuthswere determined in the usual manner[69]by elongations of close circumpolar stars,Polarisor51 Cepheibeing usually selected. The azimuth mark used was an ordinary Egyptianshamadan(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 preventany motion. The azimuths observed at the different stations are shown in the following table:—Station of Observation.Point to which Azimuth is given.Azimuth observed.Peg at West end of Muelih BasePeg east end of base33° 30′6″ E. of N.Beacon on Gebel Um HeshenibBeacon on Gebel Hamata45° 15′ 34″ E. of S.„„Hill near Gebel Selaia„„Abu Gurdi25° 30′ 35″ N. of E.„„Berenice Temple„„Kalalat61° 10′ 31″ W. of S.„„Gimeida Hill„„Hamra Dom6° 14′ 46″ E. of S.Centre of Halaib Fort„„Elba83° 25′0″ W. of S.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 formulah=dtanθ +1 −k2rd2was used, the value ofdtan θ being first found by five-figure logarithms and then that of the curvature and refraction correction1 −k2rd2by means of the very convenient “Universal” slide rule of Nessler.[71]The value ofkfound 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 correction1 −k2rd2by the slide rule, a markRwas scratched at 1210[73]on the lower scale of the slide; by bringing this markRopposite 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 ruleby 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 andhis 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.Field Number of Point.Name.Mark.Latitude N.Longitude E.Altitude Metres.°′″°′″21G. Hamrat Wogudcairn.25934342001,10329G. Iteima„2581341113849*204G. Igli„2546343616975*26aG. Atut„25056342349908*203Erf el Fahid„250534115257914G. Hagar Dungash„24591234233815*218G. Sukari„245650344250476216G. Um Tundebasummit.245548344729550*202Muelih baseE. peg.2454443457406219Isolated hill near seasummit.245423345421190*201Muelih baseW. peg.2453373441839819G. Muelihcairn.24524434037707*225Kurdeman mines„24523534413552571Marwot Rod el Ligah„21513134821514206G. Mudergegsummit.2451034221885205G. Hangalia„2450293438431,241*215G. Ghadircairn.24509344722636*32aG. Nugrus„2448343435471,505*25G. Migif„2447233427301,199*214G. Allawi„244642344939515211G. Zabara„2445213441531,361224G. Lewewi„244438344639654212Ridge near G. ZabaraN. end.2443483440561,104221Hill near seasummit.24423135315—208G. Hamrat Selma„244158342058761210Peak in G. Hafafitcairn.24404434378722240Wadi Gemal islandN. end.24404535960*223G. Sikaitcairn.24395534485771213G. Um Mogharsummit.2439163441586024aG. Abu Khrugcairn.24385734161987020G. Sufrabeacon.24384234413690241Wadi Gemal islandS. end.2438383510360207G. NahudN. cone.243536342214662*209Peak in G. Hafafitbeacon.243532344522744*222Madaret Um Gamil„243452345628454226G. Abu Hadsummit.24341234366633242Low spur of coasttip.243347351020235G. Um Regebasummit.243336344229571233G. NahudS. cone.243335342011662239Low spur of coasttip.2428433510580247G. Um Suerabsummit.2426503442331,024231G. Um Sueh„242645345430781236G. el Abiad„242627344856892*227G. Abu Hegilig„242616345832607237G. Um el Abbas„242611345633697232G. Nukheira„24255834325387673G. Um Gorafcairn.242533341828—252G. Durunkatcentre.242343344551924274S. end of low sandy island—2422413522560255G. Um SedriS. peak of twin.2422034413970234G. el Hedasummit.242056343011862248G. Sarobi (S. peak)„2420503590471*228G. Um Heshenibbeacon.2420493450531,135243G. Hefeirisummit.24202235121612246G. Khulla„241943343842978253G. TarfawiN. end of ridge.2418463455541,363251G. Marasansummit.2417343444561,261249G. Abu Ghusun„241613458171,389245G. Metawit„241551343148741*230G. Abu Hamamidbeacon.2414413447381,747250G. Um Ushersummit.2414113453211,487259G. Um Hasidok„2412453454461,497*229G. Hamatabeacon.241217350161,978258G. Abarunsummit.2411203450181,602257G. Abu Argub„241113445431,60975G. Hamrat Mukbudcairn.24953342317890254G. Ras el Kharitpeak.24925351551,661261G. Khashirsummit.24914354501,565262G. Ras el Kharitpeak.2492350281,564*256G. Kahfabeacon.248183438551,018*399Hill near Bir Qoleibbeacon.2486334246355400Bir Qoleibclay basin.24636334120239287G. el Anbatsummit.24512345458788263G. MikbiS. end of ridge.24454354491,388276G. Um Hukcairn.24441351534517279G. Um SellimE. peak.2435235951947320Peak N. of Berenicesummit.24341352630276264G. Egatpeak.24340353321,277273G. Egatsummit.24252352291,422280G. Abu Ghalqacairn.2417351647561*260G. Abu Gurdibeacon.24011355171,562288G. Derhibsummit.2408351291,160*315Hill near Berenicebeacon.2404353037196268G. el Homursummit.235854345516731289G. Um Gunudcairn.235746351210989*339Limestone hill on Ras Benasbeacon.235716354310187271G. Selaiasummit.235714345213787*317Hill on Ras Benasbeacon.235640353944193*316Limestone peak, Ras Benas„23563635369189*338Hill on Ras Benascairn.23561354053197291G. Aidabsummit.235552351338848270Hill near G. Selaia„235546345358623*275Low hill near G. Selaiabeacon.235530345437563278G. Geneinapeak near N. end.235456344745548336G. Um MaiatN. end.235445351335928*313Berenice Templebeacon.2354393528268290Marwot Elemikansummit.2354143568648319Peak N. of G. Kalalat„235410351355870334Hill near Wadi Mindeit„23535735220—340Sheikh’s tomb at Ras Benascentre.2353503547133341Tip of Ras Benassandy point.235342354750266G. Abu Deregasummit.235338345913831335G. Um MaiatS. end.235334351448842333High range N. of G. KalalatN. end.235333515187577G. el Nekeibacairn.235234342210570330Hill near Wadi Kalalatsummit.235229352218328318G. Um Maiatcentral peak.235224351536821332High range N. of G. KalalatS. end.235159351629762331G. Kalalatminor peak.235048351551743294N. end of Mukawar Island—2350473548310293G. Batogacentral peak.23507352046802329G. KalalatN. peak.234957351761,080294aS. end of Mukawar Island—2349533548190314G. Batoga (S. peak)beacon.23493735219785*282G. Kalalat„234993517361,125328G. Batogaminor peak.234857352247413353Minor peak of G. KalalatS. one of pair.234737351846894355G. Um Hegiligsummit.234713351432966327G. DibagNW. peak.234647352123517308G. DibagSE. peak.23464035224544*265G. Dahanibbeacon.2345443511101,270*269G. Um Bisella„234534345739824*277G. Zergat Naam„234528344034823345G. ReyanN. peak.234520351750740346G. ReyanS. peak.23447351720863*343Hill near Shenshef ruinsbeacon.23445352240290292Erf el GemalW. end of ridge.234337345210673272G. Hagar el Filsummit.2343534423845281G. Shut„234235351659930398Bir Abu Hashimprincipal well.23415634426320*397Hill near Bir Abu Hashimbeacon.23414434333386348Hill near W. Salib Abiadsummit.234153573703325G. Um Etlicentral peak.233954352322795384G. Abu Husenatsummit.23394535115725326G. Um EtliW. peak.233932352152844324G. Um EtliE. peak.233926352353764295G. Hendusisharp peak.233915345844678344G. Um Akrabeacon.2337583516411,050*385Ruins of Um Eleiga„2337203535599302Peak near G. Abu Dahrsummit.23371635644840*79Um Harbabeacon.233656343038688356G. Um AkraS. peak.233635351658970347G. Hindiasummit.233627351339873337Zeberged Islandcentral peak.233616361142238*267aG. Abu Dahrbeacon.23368355461,131305G. Abu Sieiyilsummit.2335163514833321G. Faraidpeak.233483523181,131323G. Faraid„2333293521501,341361Hill S. of G. Abu Dahrsummit.23332835442912322G. Faraidhighest peak.233373522101,366342G. Faraid„2331533522461,068360Hill S. of G. Abu Dahrsummit.23314335515772362Hill S. of G. Abu Dahr„23313735249793359Hill S. of G. Abu Dahr„23311735622784*285G. Faraidpeak.2330533520251,259*85G. Um Khafurbeacon.232954342919560357G. Faraidminor peak.232935351730862286G. Faraid“The Bodkin.”2328583520351,232350G. Faraidoutlying peak.232839351735875351G. Faraidpeak.232835351631954352Granite peak near Wadi Rahabasummit.232748351315697349G. Faraidoutlying peak.23271435195904297G. Orgabeacon.23262135817682363Hill near Wadi Rahabasummit.232618351238506*296G. Abraqbeacon.232519344648667365Hill near Wadi Rahabasummit.23252351142536389G. Faraidminor peak.232457352442585358G. Faraidpeak.232239352219916395aG. Abraq„232226345018699395G. Abraq„23221934504705*87G. Awamtibbeacon.232059342639793*299G. Um Tenedba„231948351040656368Hill near G. Um Tenedbasummit.231843351236555*390Plateau near Abu Saafabeacon.231840344830639307G. Saalekpeak E. side.231818343110753*354aG. Fereyidbeacon.231729352248612394G. Hodein„231620345325695380G. Hodeincorner of scarp.23161834525718304Peak W. of G. Um Reitsummit.231526343014805306Peak W. of G. Um Reit„231523343147837366Hill near Wadi Rahaba„23154351625420*300G. Um Reitbeacon.23154343417857*298aG. Harhagit„231435351252542393G. Tibatibsummit.23124135128396387Hill near Wadi Hodein„231224351651383396Granite peak near Wadi Rahaba„231111352035355392Rock in sea(level datum).2310553535570310Isolated hill west of G. Etresiasummit.231035342523703388Hill near Wadi Hodein„231033351632309423G. Kala„23103344549808503G. EtresiaE. summit.239243430431,037303Hill E. of G. Etresiasummit.23915343223922*375aG. Anfeibbeacon.23824345919705364Hill NE. of G. el Anbatsummit.2380352146359*386Close to Bir Shalateinbeacon.2375735362313502G. Etresiasummit.237533431241,038407G. Kala„23742344258846311Isolated hill north of G. Shigigat„23716342439901425G. Kala„23711344237783424G. Kala„23651344230739501G. Etus„23619342953997*367aG. el Anbatbeacon.2365351927390312G. Shigigatconspicuous peak.235373423391,023406G. Kalasummit.23516344219629500G. Aqab el Negum„233263427201,149496G. Natetiaipeak.2338342101,022499Peak near Aqab el Negumsummit.2324334263998498Peak near Aqab el Negum„23222342536979497Peak near Aqab el Negum„23156342542974495G. Natetiaipeak.23049342271,164373G. Niqrub el Tahtanihigh pinnacle.2304235058828373aG. Niqrub el Tahtanibeacon.2303935053829*369aG. Beida„23014351654716494G. Natetiaipeak.225935342222977508Peak in S. part of G. Fegsummit.225845342029885391G. Humariai„22583835936563382Hill S. of Wadi Beida„225810351258631507Peak in S. part of G. Feg„22582341915836381aHill S. of Wadi Beida„225730351653631379G. Mismih„225656344532599381Hill S. of Wadi Beida„22564735173615510Conical hill„225448344027504*401G. Kolaiqobeacon.225413352435320*374G. Niqrub el Foqani„2251293456491,078509G. Waqifsummit.225121343857556466Tree at Mersa Shabcentre.225056354731419G. Um el KalalaN. peak.225023344414672506G. Sheyenit„225013341823853505G. SheyenitS. peak.224958341815887415G. Um el Kalala„224926344418655418Hill close to Bir Madibeacon.22474635139556*408G. Meneiga„224735351171,032*402G. Meneigahighest point.2247313510571,092430Hill near Wadi Tikoshasummit.224629352011443*410Gemeida hillbeacon.224629353749123437Hill SSE. of Bir Meneigasummit.22455835133987434Peak near Wadi Ti Ilak„224436351754774376G. Shabihhighest peak.2244263450211,117527G. MishbihNW. summit.2244213441231,321378G. Mishbihbeacon.2244183441201,316440Ridge near head of W. Shellal el Gharbihigh point.2244153512181,092447Qrein Salamasummit.22445352432354435Peak near Wadi Ti Ilak„22444351746849431Hill N. of Wadi Muqur„224355351852670414G. el Nagasmaller hill.22433234272274791G. Seigacairn.224331341616905377G. MishbihE. peak.2243283442391,311526G. Mishbihpeak.2242523441271,353413G. el NagaN. peak.224242342746787445G. Tueiwisummit.22424035556836*403G. Gerfhigh point on ridge2242153511431,339427G. Gerfminor summit.2242133513321,327*370G. Gerfbeacon.224263512161,419412G. el NagaS. peak.22420342825827436Peak near Wadi Muqursummit.224158351822778493G. MishbihS. peak.224144344241988432Peak near Wadi Muqursummit.224140351735962383G. Muqur„2241313516501,058433Hill near Bir Sararat Seyet„22411635057797426Peak near Wadi Diqdibsummit.224163514301,258438Peak near Wadi Qadiloi„22413351754911429Peak near Wadi Eirahimib„224051359551,227371G. Shweib„224043344357914428Peak near Wadi Eirahimib„2240383510291,327531Mt. near Bir Baaneit„22401351815909404G. Gerfminor summit.223959351071,318405G. Korabkansi„223939350441,052446G. KorabkansiN. peak.2239353459321,176614G. Hamra Dom„223934353819326439Peak near Wadi Diqdibsummit.2239173516111,028*372G. Korabkansibeacon.2239163459551,230420Peak near Wadi Difotebcairn.2239113510211,298*417G. Hamra Dombeacon.22396353842388450Tibansi Tikam Ankweisummit.22395353211345475Peak near Wadi Sherefa„2238203511421,344411G. Hamra Domcentral peak22382353912381473Hill near Wadi Um Sahasummit.223754351756926532Hill near Wadi Um Saha„223753351756932444G. Drebpeak.223654356201,139615G. Hamra DomS. peak.223648353936317443G. Drebpeak.22364135701,148442G. Dreb„223640357461,137476G. Dreb„223616359151,176478G. Dreb„223610358211,224477G. Dreb„22361358561,191504G. Kulyeitsummit.223550341638724441aG. Drebpeak.223518358241,293533Kilia Aribsummit.22352351846647441G. Drebpeak.22350358361,288535G. Dreb„223415359421,095534Tibashoi Tomakolatsummit.22334351940464*449aKolmanab hillbeacon.223226355336137566Hill near Wadi Ibibsummit.22322353736459619Point near Kolmanab—223155355240110612G. Medaraiminor peak.223123351351,103520G. Medaraisummit.223058351251,299610G. Medaraiminor peak.223056351437922485G. AnweiyibN. summit.223022345342871*567aG. Eqrun (W. summit)beacon.22301035379473567G. EqrunSE. summit.22300353718468665Adar Aweib Um Bishtitsummit.22292835332455577Granite hill„22291935546143522G. MedaraiS. peak.22298351241,114528G. Anweiyibsummit.22295345332864587G. Um RaseinN. peak.222852352044791*416G. Um Raseinbeacon.222812352019909*575Einiwai hill„222750355759138521G. Abu HireiqN. summit.2227393514581,116609Hill near Wadi Merdiyebsummit.222721351551856613Tahaqayet„222647354023432596Hill near Wadi Odruk„222647351842639595Hill near Wadi Odruk„222625351917615582Granite hill„222623355620155592Near Wadi Odrukpeak.2226935209627523G. Abu Hireiqhighest point.2225573514391,319581Osnei hillsummit.222538355042251455Hill near Bir Um Bishtit„222514353319634472Mt. near Wadi Abu Hireiq„2224433516131,256611G. Adatalob Hadalbeacon.222438354822381451Titailibabsummit.222432353852593459G. OrgemNW. peak22243135311779576G. OrgemSE. peak.222421353111775488G. AnweiyibNW. summit.222416344946884471Mt. near Wadi Abu Hireiqsummit.2223563516411,226486G. AnweiyibSE. summit.222337345125921664Hill S. of Titailibabsummit.22232635395554422G. Abu Hodeidbeacon.222318351491,482572G. Adatalob Adarasummit.22231354958385603Near Wadi Tikraneibpeak.222257351838732454G. Meis-heit-arsummit.222237353444717470Mt. near Bir Abu Hodeid„2222313517331,074479G. Mansur Diab„2222283510491,091469Mt. near Bir Abu Hodeid„2222103517511,078457Hill near head of Wadi Qidmib„22223353242738583G. Tishushi Tibokiminor summit.22220355712284647Abu Hodeid Oqlasummit.222129351751992492G. Um el Tiur el Tahtani„222126343446783666G. Meis-heit-ar„222115353448721580G. Tishushi Tibokihighest point.222114355546359*569aTi Keferiaibeacon.22210354947494598G. Hamra TitSW. peak.22206352044642461G. Qidmibsummit.2219553530451,037*456O Shakafa„221944353442751597G. GerorNE. summit.22193135524434462G. Qidmibsummit.2219133530231,108464G. Qidmib„221943529181,070463G. Qidmib„2218493530151,099616G. Gerorhigh point.221837355018510584G. Qidmibsummit.2218183529201,089453Adar Aqdeibsummit.22186353751736490G. Um el Tiur el Foqani„221756344114946602G. Gerorsummit.221751355055516649G. el Selapeak.221737361417433487G. Hadal Derqasummit.221640359551,108*460Hadal Aweib Meisahbeacon.2216393531551,224650G. el Selapeak.221632361259560491G. el Adraqsummit.22162234353770421Qara Saba„22167354134778601G. Um Seleim„2215583521351,098480G. Hadal DerqaSE. peak.2215503510591,090585Hadal Aweib Meisahminor summit.2215483530101,092604G. Leqaqsummit.221540352081,192512G. HeianaiNW. summit.221440351301,007668G. Hamidaminor summit.221434354613583620Qash Amirbeacon.221431361220724651G. Sul Hamidhighest peak.22141636453572568G. Balatitdasummit.22135635585592513G. Hilwit HasiumNE. peak.2213503514381,037468G. Adar Qaqasummit.221347351901,606482G. HeianaiSE. summit.221341353111,256618G. Hamidaminor summit.221336354647701608G. Adar Qaqa„2213333519211,542448G. Hamidasummit.221332354633754*458Halaib Fortcentre of top.2213253638568556G. Elbapeak near Bir Kansisrob.221321362223820652Hill E. of G. Balatitdapeak.221320355943491555Karam Elbasummit.221313362540586525G. Shanaiyet„22134344943907653Hill near Wadi Warabeit„22134355012511676G. Elbapeak near Wadi Yahameib.22132362054935662G. Balatitdaminor summit.221256355744493489G. el Hateibsummit.221244344318854674Hill near Wadi Siamtitsummit.221242361310507516G. Hilwit HasiumSW. peak.221230351314952669Mt. south of Hadal Aweib Meisahsummit.2212213530401,072481G. el Arib„221215358321,112579aG. Um Einbeacon.22115235394901667Hill near head of Wadi Warabeitsummit.22114035476651467G. Soaorib„2211333520131,469*536G. Elbabeacon.2211273620521,428621G. Elbaminor peak.2211163623441,102675G. Elba„2211103620351,394483G. Heleikontisummit.221173457591,151573G. Warabeit„22111354752794518G. Soaorib„2210543517481,397*571Adar Aweibbeacon.22105035540620538G. Elbaminor peak.2210173619341,217606G. Soaoribsummit.2210143521251,34998G. Muqsimcairn.22101134112825589Mt. near Wadi Baueiwaisummit.2210103530451,266537G. Elbahighest point.221033621521,435617G. Mashushenaisummit.22954354945634605G. Soaoribpeak.229503521451,383678G. O Sir Eirabsummit.22924362056842677Hill near W. O Sir Hadal„2297361556724519Peak near Wadi Kirir„228243518271,328465G. Soaorib„228223523361,431590Mt. N. of Wadi Adar Ameit el SharqiNE. peak.22816353121,294670Hill near Wadi Aqwemsummit.22812354330724578Hill near Da-aiyob Wushaq„2274535448833681G. Hanqufminor peak.22745361654789591Mt. N. of Wadi Adar Ameit el SharqiSW. peak.227123529541,299682G. Hanqufminor peak.22654361718877524G. Um Reddamsummit.226233459551,109657G. MiatitW. peak.226143534121,229511Mt. near G. Egatsummit2255334531985484G. Egat„225393452151,145586G. MiatitSE. peak.225363535341,257659Mt. near Wadi Adar Ameit el Gharbipeak.225203525301,440540G. HanqufN. peak.224523618451,397452Hill near W. Di-ibsummit.2241436132491655G. SurukE. peak.22410353871,059588G. Surukhighest point.223503535281,327673Hill near G. Surukpeak.22346354016846656G. Shendib„22340353611,275515Mt. near G. Himeitra„223173517521,134685G. Shendib„22383613441,427539G. Shendodaihighest peak.22313625311,529672Hill near G. Surukpeak.2230354025847661G. Shiab„22257354518855*607G. Hadarbabeacon.22253364723217654G. Shiabhighest point.2225135447987517G. Himeitrasummit.222463514191,231474G. Iscairn on summit.22236352841,736599G. Ispeak.222243527351,659627G. ShendodaiS. peak.222123625221,395684Hill near G. Shendibsummit.222123696601600G. Ispeak.222113526561,594671Hill near G. Suruk„22154354020903593Mt. east of G. IsN. peak.221383531561,290570G. Hanqufhighest point.221323620141,465640G. Shendibpeak.221263614471,724641G. Shendib„22183614591,698594Mt. east of G. IsS. peak.22123531111,474642G. Shendibpeak.22051361501,696543G. Shendibbeacon.220483616301,912546G. Shendibpeak.220463614491,674544G. Shendibpeak.220453616101,852687Hill near G. Shellalsummit.220403692614557G. Shellalpeak.220393630301,269529G. ShellalE. peak.220343630451,279574G. Shellalhighest point.220153629401,409545G. Shendibpeak.215956361761,863547G. Shendib„2159493615421,668550G. Shendib„215983614161,565645G. Shendib„215973613221,227549G. Shendib„215963614401,525644G. Shendib„215903615201,552646G. Shendib„2158583613111,196660Low hill near W. Shendibsummit.21582436425500541G. QedaE. peak.2156243628111,850542G. QedaW. peak.2156163626121,872514Hadal Aweibsummit.2152403522261,780628G. Asotribahighest point.2151553630262,216551Eir Aweitpeak.2150303622481,715552Eir Aweit„214956362311,678564G. ObkeikN. peak.2149553539401,837548G. AsotribaS. peak.2149303630402,082565G. ObkeikS. peak.214963539291,856553Eir Aweithighest peak.2147573622401,853554Eir Aweitpeak.2147383622321,759658Adar Itsharp peak.214713619141,170409beacon.214421343653—558Aritpeak.2140263622501,529559Arit„2139253622321,772663Karai Awasummit.2139183610211,056562Aritpeak.213833618571,410560Arithighest peak.2136513623271,810563Arithigh peak.213613619521,727561AritW. end of ridge.2135503622481,532
35·8 + 1·000dφ− 0·011dT−h0= 0(1)
35·8 + 1·000dφ− 0·011dT−h0= 0(1)
35·8 + 1·000dφ− 0·011dT−h0= 0(1)
αColumbæ.
Whence the equation for αColumbæis
51·1 − 0·896dφ+ 0·404dT−h0= 0(2)
51·1 − 0·896dφ+ 0·404dT−h0= 0(2)
51·1 − 0·896dφ+ 0·404dT−h0= 0(2)
εCanis majoris.
Whence the equation for εCanis majorisis
51·5 − 0·801dφ+ 0·546dT−h0= 0(3)
51·5 − 0·801dφ+ 0·546dT−h0= 0(3)
51·5 − 0·801dφ+ 0·546dT−h0= 0(3)
Collecting the equations of the three stars, we have
35·8 + 1·000dφ− 0·011dT−h0= 0(1)51·1 − 0·896dφ+ 0·404dT−h0= 0(2)51·5 − 0·801dφ+ 0·546dT−h0= 0(3)
35·8 + 1·000dφ− 0·011dT−h0= 0(1)51·1 − 0·896dφ+ 0·404dT−h0= 0(2)51·5 − 0·801dφ+ 0·546dT−h0= 0(3)
35·8 + 1·000dφ− 0·011dT−h0= 0(1)
51·1 − 0·896dφ+ 0·404dT−h0= 0(2)
51·5 − 0·801dφ+ 0·546dT−h0= 0(3)
By solving these equations fordφ, we find
dφ= + 6″·5
dφ= + 6″·5
dφ= + 6″·5
whence the latitude is found to be
24° 20′ 56″·5
24° 20′ 56″·5
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 ofPolarisat 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 whenPolarisis 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.
Azimuthswere determined in the usual manner[69]by elongations of close circumpolar stars,Polarisor51 Cepheibeing usually selected. The azimuth mark used was an ordinary Egyptianshamadan(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 preventany 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 formulah=dtanθ +1 −k2rd2was used, the value ofdtan θ being first found by five-figure logarithms and then that of the curvature and refraction correction1 −k2rd2by means of the very convenient “Universal” slide rule of Nessler.[71]The value ofkfound 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 correction1 −k2rd2by the slide rule, a markRwas scratched at 1210[73]on the lower scale of the slide; by bringing this markRopposite 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 ruleby 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 andhis 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.
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.