NOTES ON THE GEODETIC SURVEY OF SOUTHERN RHODESIA

Abstract

Triangulation.—Apart from Simms' Geodetic Chain, Gordon's Chain, the Copper Queen Limb, and a section of the Victoria and Umtali Series, all the primary triangulation shown on the accompanying map has been executed since 1933. The work of Simms and Gordon has been remodelled, however, being greatly strengthened, and these chains are now called Simms' and Gordon's Series. For an explanation and plan of the above Series, see “A Note on the Trigonometrical Survey of S. Rhodesia”, in the Empire Survey Review, no. 27, vol. iv.     

THE TRANSVAAL-SOUTHERN RHODESIA GEODETIC CONNEXION

Abstract

In the original geodetic series in Southern Rhodesia—completed by Mr Alexander Simms in 1901—the geographical coordinates of all stations were referred to the point SALISBURYas origin. The coordinates of SALISBURY were fixed by interchange of telegraphic signals with the Royal Observatory at the Cape for longitude, combined with astronomical determinations of time, latitude, and azimuth (see Vol. III, “Geodetic Survey of South Africa”).     

THE GEODETIC SURVEY OF CANADA

Abstract

Shortly after the inception of the Geodetic Survey of Canada in 1905, reconnaissance for primary triangulation was commenced in the Ottawa-Montreal area. About the same time, precise levelliilg operations were begun from a bench mark already established by the United States Coast and Geodetic Survey near the International border at Rouses Point in Quebec.     

PRACTICAL APPLICATION OF THE LAPLACE LONGITUDE-AZIMUTH RELATION TO CONTROL OF GEODETIC ANOMALIES

Abstract

1. In geodetic work a ‘Laplace Point’ connotes a place where both longitude and azimuth have been observed astronomically. Geodetic surveys emanate from an “origin” O, whose coordinates are derived from astronomical observations: and positions of any other points embraced by the survey can be calculated on the basis of an assumed figure of reference which in practice is a spheroid formed by the revolution of an ellipse about its minor axis. The coordinates (latitude = ?, longitude = λ and azimuth = A) so computed are designated “geodetic”.     

GEOGRAPHICAL POSITIONS IN MALAYA AND SIAM

Abstract

Malaya.—The geographical positions of points in the “Primary Triangulation of Malaya”, published in 1917, depend upon latitude and azimuth determinations at Bukit Asa and on the longitude of Fort Cornwallis Flagstaff, Penang, the latter being supposed to be 100° 20′ 44″.4 E. This value was obtained by Commander (later Admiral) Mostyn Field in H.M.S. Egeria 1893, by the exchange of telegraphic signals with Mr Angus Sutherland at Singapore, Old Transit Circle. The longitude, 103° 51′ 15″.75 E., accepted for Singa- pore in order to arrive at this determination of Fort Cornwallis Flagstaff, was based upon that of an Observation Spot, 103° 51′ 15″.00 E., fixed in 1881 by Lieutenant Commander Green, United States Navy, by meridian distance from the transit circle ofMadras Observatory, the corresponding longitude of the latter being taken as 80° 14′ 51″.51 E.     

THE NEW GEODETIC TAVISTOCK THEODOLITE

Abstract

Work on the original Geodetic Tavistock Theodolite was commenced in the autumn of 1931, and after suitable tests this instrument was sent out to East Africa and used on the East African Arc. Bt Major M. Hotine, R.E., writing in the E.S.R. of April 1935 (no. 16, vol. iii), stated: “The Tavistock instrument, although a first model, gave uniformly satisfactory service throughout and was used for over half the main angular observations.”     

TRIGONOMETRICAL LEVELLING ACROSS THE INLAND ICE IN NORTH GREENLAND

Abstract

As part of the scientific work of the British North Greenland Expedition (1952–1954), a programme of trigonometrical levelling was carried out from the east to the west coast of Greenland, along a line across the inland ice between latitudes 76° 40′ N., and 78° 10′ N. The primary purpose of the work was to determine accurately the heights above sea level of a series of gravity stations, the gravity measurements being made in connection with determinations of ice thickness. For meteorological purposes it was necessary to know also the altitude of the Expedition's central station, situated in latitude 78° 04′ N., longitude 38° 29′ W. The accuracy necessary for the purpose of the gravity survey was a few metres for the altitudes, while the latitude of each gravity station had to be determined with an accuracy of ± 0.1 minute.     

LONG LINES ON THE EARTH: THE DIRECT PROBLEM

Abstract

19. Formulae.—In Nos. 6, vol. i, and 9, vol. ii, pp. 259 and 156, there has been described a new method for dealing with long geodesics on the earth's surface. There the so-called “inverse” problem has claimed first attention: given the latitudes and longitudes of the extremities of a geodesic, to find its length and terminal azimuths. It remains to discuss the “direct” problem : a geodesic of given length starts on a given azimuth from a station of known latitude and longitude; to find the latitude and longitude of its extremity and the azimuth thereat. The solution of this direct problem demands a certain recasting of the formulae previously given. In order of working the several expressions now assume the forms below.     

NOTES ON THE BASE LINES OF THE CEYLON TRIANGULATION

Abstract

The triangulation of Ceylon depends for its scale upon two bases, each about 5½ miles long, situated at Negombo on the West Coast (latitude 7° 10′) and at Batticaloa on the East Coast (latitude 7° 40′). Both bases are in low, flat country; brick towers up to 70 feet high had to be built over the terminals to enable observations to be taken to surrounding points. These lines have recently been re-measured.     

NOTES ON THE USE OF ANEROID BAROMETERS IN THE GOLD COAST

Abstract

For several years it had been realized that aneroids in the Gold Coast showed a distinct lag in the readings when subject to fairly large changes of height. The range of height in the Colony, however, being relatively small and control heights fairly numerous, little interest was taken in the cause, which was generally thought to be due to hysteresis. * All aneroids in use on the Gold Coast are graduated on Airy's scale which is based on latitude 45° and temperature 50° F. In 1921 Mr. C. L. T. Griffith, at that time Chief Instructor of the Survey School, carried out various tests with a number of aneroids, and from these tests concluded that the main source of error arose from inappropriate graduation of the height-scale relative to pressure; using as constants latitude 15°, temperature 86° F., and mean humidity 67 per cent., he worked out a proposed general scale for the Tropics. Ten years later the purchase for test purposes of new aneroids graduated to this scale was considered but was eventually postponed when it was learnt that the question of a special scale for use in the Tropics was under consideration at home by a special Committee consisting of representatives of the Admiralty, War Office, Air Ministry, and National Physical Laboratory.     

SIR DAVID GILL AND THE GEODETIC SURVEY OF SOUTH AFRICA AND THE ARC OF THE 30TH MERIDIAN

Abstract

In the memoir of the late Capt. G. T. McCaw which appeared in the January number of this Review (vii, 47,2), reference was made to the part which the late Sir David Gill played in the origin of the work on the survey of the Arc of the 30th Meridian in Africa. This year is the centenary of Gill's birth, as he was born in June 1843, and it is therefore timely to give some account of his work during his long term of office as Her Majesty's Astronomer at the Cape which resulted inthe inception and completion of the Geodetic Survey of South Africa and the survey of the Arc to the southern shores of Lake Tanganyika. He died on 24th January 1914.     

MAPPING OUT OF CROWN LANDS IN CEYLON

Abstract

The Island of Ceylon has an extent of 25,332 square miles, and a population of nearly seven millions; the range of latitude is from 5° 55′ to 9° 50′ North and of longitude 79° 42′ to 81° 53′ East.     

Toute la Geodesie sans ellipsoide Les reperes laplaciens Application plus particuliere aux travaux a latitude elevee

Resume Après de nombreuses années d’hésitation, on a finalement reconnu, au Congrès de Florence, en 1955, que dans le repérage des altitudes, seule la notion depotentiel était claire et sans ambigu?té, l’altitude au sens courant du terme étant conventionnelle. De la même fa?on, pour le repérage géométrique des points à la surface de la Terre, les coordonnées (X Y Z) des points, dans letrièdre cartésien terrestre général, sont les inconnues fondamentales; les coordonnées géodésiques couramment utilisées (longitude, latitude altitude H au-dessus de l’ellipso?de) sont conventionnelles. Mais pratiquement, afin d’écrire commodément les relations d’observation, il para?t intéressant de passer par l’intermédiaire detrièdres locaux (trièdres laplaciens), liés de fa?on invariable au système cartésien général, et de repérer toutes les grandeurs dans ces trièdres locaux. Toutes les observations utilisées en Géodésie s’expriment de fa?on simple et sans singularités dans ces trièdres locaux. La jonction des triangulations classiques, l’Astrogéodésie, la synthèse des Géodésies classique et spatiale sont facilitées. En astronomie de position, les grandeurs longitude, latitude, azimut, sont avantageusement remplacées par: déviation Est-Ouest, déviation Nord-Sud, azimut de Laplace. Les relations d’observation s’écrivent sans difficulté, même dans les régions polaires. L’application pratique des nouvelles formules obtenues a été réalisée avec succès par L.F. Gregerson (Service Géodésique du Canada).

---

Summary At Florence, in 1955, it was accepted that, in the problems of levelling, the notion ofpotential was scientifically clear, and that the altitude could derive from it only through a conventional process. In the same manner, when we want to have a geometric reference of the points at the earth surface, we use the coordinates (X Y Z) in thegeneral cartesian trihedron as fundamental unknowns, the geodetic coordinates (λϕH) deriving from (X Y Z) through a conventional process. Practically, in order to set up the observation equations, it is necessary to define local trihedrons (laplacian trihedrons), deriving from the cartesian general system through a fixed transformation, and to refer all the unknowns in these local trihedrons. All the observations used in Geodesy can be expressed simply and without any singularity in these local trihedrons. The links between classical geodetic nets, the astrogeodesy, the combination between classical and spatial geodesy, become easier. In astronomical controls, “longitude, latitude, azimut” must be replaced by: W-E deflection, N-S deflection and Laplace azimuth. Thus all the observation equations can be set, even in polar regions. A practical application of the new formulae was done successfully by L.F. Gregerson (Geodetic Survey of Canada).

     

GEOGRAPHICAL POSITIONS IN MAURITIUS AND RODRIGUEZ

Abstract

The desirability of determining as accurately as possible the latitudes and longitudes of a number of points in Mauritius appears to have first appealed to the French astronomer M. L'Abbé de la Caille, who was sent to the island by the French East India Company in 1753. His observatory was situated 4,730 feet east and 2,610 feet north of Port Louis Time Ball, and in addition to determining the geographical position of this. 0bservatory (the house, later demolished, of a Mr Mabile, where Mr D'Après had made observations the previous year)—

Latitude 20° 09′ 42″ S.; Longitude 55° 08′ 15″ E. of Paris—,

he succeeded in effecting a triangulation of the island. Four bases were measured with wooden scales which previously had been compared with an iron toise (6.394 feet—the French fathom) approved by the Academy of Sciences, Paris; a I4-inch quadrant fitted with micrometer was used to measure the angles.     

Iterative vector methods for computing geodetic latitude and height from rectangular coordinates

 Two iterative vector methods for computing geodetic coordinates (φ, h) from rectangular coordinates (x, y, z) are presented. The methods are conceptually simple, work without modification at any latitude and are easy to program. Geodetic latitude and height can be calculated to acceptable precision in one iteration over the height range from −10⁶ to +10⁹ m. Received: 13 December 2000 / Accepted: 13 July 2001     

ASTRO FIX BY TWO EX-MERIDIAN STARS

Abstract

DR. DE GRAAFF-HUNTER proposed two new astronomical methods in a paper which he read at the Conference of Commonwealth Survey Officers, and the writer recently had an opportunity of trying out one' of these, with some interesting results. The method used, which requires timed intersections on a pair of stars in azimuths differingby about 90° and depends upon the alg'ebraic solution of the pair of position lines so formed*, will yield latitude, longitude and azimuth. The observations are brief and uncomplicated, prior identification unnecessary",and ~4e subsequent. computation is light: and requires no more than about thirty minutes for a pair of stars, inclusive of star identification.     

THE TRANSVERSE MERCATOR PROJECTION

Abstract

I. Introduction.—Map projection is a branch of applied mathematics which owes much to J. H. Lambert (v. this Review, i, 2, 91). In his “Beyträge zum Gebrauche der Mathematik und deren Anwendung” (Berlin, 1772) he arrived at a form of projection whereof the Transverse Mercator is a special case, and pointed out that this special case is adapted to a country of great extent in latitude but of small longitudinal width. Germain (“Traité des Projections”, Paris, 1865) described it as the Projection cylindrique orthomorphe de Lambert, but he also introduced the name Projection de Mercator transverse or renversée; he shows that Lambert's treatment of the projection was remarkably simple.     

Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

We present software for spherical harmonic analysis (SHA) and spherical harmonic synthesis (SHS), which can be used for essentially arbitrary degrees and all co-latitudes in the interval (0°, 180°). The routines use extended-range floating-point arithmetic, in particular for the computation of the associated Legendre functions. The price to be paid is an increased computation time; for degree 3,000, the extended-range arithmetic SHS program takes 49 times longer than its standard arithmetic counterpart. The extended-range SHS and SHA routines allow us to test existing routines for SHA and SHS. A comparison with the publicly available SHS routine GEOGFG18 by Wenzel and HARMONIC SYNTH by Holmes and Pavlis confirms what is known about the stability of these programs. GEOGFG18 gives errors <1 mm for latitudes [-89°57.5′, 89°57.5′] and maximum degree 1,800. Higher degrees significantly limit the range of acceptable latitudes for a given accuracy. HARMONIC SYNTH gives good results up to degree 2,700 for almost the whole latitude range. The errors increase towards the North pole and exceed 1 mm at latitude 82° for degree 2,700. For a maximum degree 3,000, HARMONIC SYNTH produces errors exceeding 1 mm at latitudes of about 60°, whereas GEOGFG18 is limited to latitudes below 45°. Further extending the latitudinal band towards the poles may produce errors of several metres for both programs. A SHA of a uniform random signal on the sphere shows significant errors beyond degree 1,700 for the SHA program SHA by Heck and Seitz.     

THE RE-TRIANGULATION OF GREAT BRITAIN-II

Abstract

Reconnaissance.—It has been possible to draw up a “paper scheme” for most of the primary triangulation by examination of large-scale topographic maps for possible obstructions to the proposed rays (after due allowance for curvature and refraction along the ray); the fact that certain of the proposed lines had been definitely observed in the existing primary or secondary triangulation was of course of material assistance. Since, however, all observations were to be to luminous beacons, requiring a close organization, it would have been unsound to draw up an observing programme on the strength of this paper scheme alone. The omission of a few key rays, subsequently found to be obstructed by timber, whose height cannot be appreciated from maps, or by local features which had grown up since the last triangulation, would have entailed some confusion in the observing programme, the establishment of additional stations for occupation during a later season, and the reoccupation of stations surrounding such additional points; in short, the probability of a year's delay in obtaining sufficient material for adjustment, and a considerable loss of economy. Arrangements were accordingly made for the paper scheme of the English main chain to be verified and amended on the ground by special reconnaissance parties in 1935, when in fact no instruments or beacons were available to commence observing in any case. In the same way, field reconnaissance of the Scottish main chain and of the Western figure (Wales and the S.W. peninsula) was completed in 1936 and a start made on the Eastern figure (East Anglia and S.E. England), so as to get the reconnaissance and station preparation well ahead of observing. Experimental reconnnaissance of a few secondary blocks—for which, as also for the primary reconnaissance of the flat enclosed East Anglian country, paper schemes are practically useless—was also commenced in 1936 and is being pushed ahead rapidly in 1937.     

TRANSVERSE MERCATOR FORMULAE

Abstract

The Transverse Mercator Projection, now in use for the new O.S. triangulation and mapping of Great Britain, has been the subject of several recent articles in the “Empire Surpey Review. The formulae of the projection itself have been given by various writers, from Gauss, Schreiber and Jordan to Hristow, Tardi, Lee, Hotine and others—not, it is to be regretted, with complete agreement, in all cases. For the purpose for which these formulae have hitherto been employed, in zones of restricted width and in relatively low latitudes, the completeness with which they were given was adequate, and the omission of certain smaller terms, in the fourth and higher powers of the eccentricity, was of no practical importance. In the case of the British grid, however, we have to cover a zone which must be considered as having a total width of some ten to twelve degrees of longitude at least, and extending to latitude 61 °north. This means, firstly, that terms which have as their initial co-efficients the fourth and sixth powers of the longitude ω (or of y) will be of greater magnitude than usual, and secondly that tan² ? and tan⁴ ? are likewise greatly increased. Lastly, an inspection of the formulae (as hitherto available) shows a definite tendency for the numerical co-efficients of terms to increase as the terms themselves decrease—e.g. terms in η⁴, η⁶, etc.