THE SCALE ERRORS OF AIRY'S PROJECTION BY BALANCE OF ERRORS,WITH SPECIAL REFERENCE TO THE BRITISH ISLES

Abstract

THE formula for the projection is based upon the spherical assumption. To calculate it for the spheroid might be very complicated and would not be worth while. The projection is suitable for very large areas as a compromise between the Zenithal Equal-area projection on the one hand and the Zenithal Equidistant or Zenithal Orthomorphic on the other. Its application to an area as small as the British Isles would not serve any useful purpose. An analysis of its errors in the general case reveals some unexpected simplicities. This analysis is given below, followed by its application to the particular case of the British Isles on the ten-mile scale. This is done merely to find out what changes would have occurred if the supposed drawing of that map on Airy's projection had been real.     

THE TRANSVERSE MERCATOR PROJECTION: A CRITICAL EXAMINATION

Abstract

In January 1938 the writer decided against holding up for more years some work on the Transverse Mercator Projection (E.S.R., 27, 275). The extension to the spheroid was not then complete, nor is the present paper to be regarded as a logical continuance. It is first proposed to show the results of “transplanting” orthomorphically upon the spheroid a spherical configuration forming a graticule.     

MAP REPRODUCTION IN THE TROPICS

Abstract

The useful paper on “Map Printing in Ceylon” which was written by Mr. W. S. Maddams of the Ceylon Survey Department for the 1935 Conference of Empire Survey Officers has been read with interest in the Gold Coast. The writer of the present note was not present at the Conference but has read the report of the discussion on that paper. It appears to him that Mr. Maddams makes certain generalizations, based on his own, experience in Ceylon, regarding map reproduction in the Tropics and, however true these statements may be as regards Ceylon, they certainly do not apply to the Gold Coast and probably not to some other tropical countries. As it is possible that another Colony, when starting a map-printing establishment of its own, might be led, on the strength of the paper read at the Conference, to believe that less expensive (though less elaborate) methods than those used in Ceylon are not possible in a tropical country, the following notes on the methods used and found successful in the Gold Coast may be of some value. They are not intended to be a criticism of Mr. Maddams' paper but only to supplement it with a record of experience obtained in another part of the world.     

INFLUENCE OF GEOIDAL TILT ON AZIMUTH

Abstract

If the horizontal circle of a theodolite is tilted, the azimuth of a point in the field will be affected. The tilt may be a mere question of dislevelment but it may be also due to a deviation of the vertical which the level on the instrument cannot measure. In either case the effect on azimuth or bearing is of the same nature, which is worth consideration for more than one reason. That it has previously been investigated elsewhere is no valid reason against renewed consideration here, particularly as the consequences are probably unknown to some readers.     

THE GEODETIC EVIDENCE CONCERNING WEGENER'S HYPOTHESIS

Abstract

Twenty years ago there seemed to be many good reasons for supposing that Wegener's hypothesis of Continental Drift was untenable. There seemed to be unsurmountable objections to it from a point of view of a mechanism capable of causing tectonic movements of such magnitude; there was disagreement over the correspondence of the orogenic belts in the areas of the earth which are supposed formerly to have been united, as well as over other major geological factors; there was the gravest discordance between the known palaeontological facts and those demanded by the theory; after examination of the geodetic evidence, at least one distinguished geodesist pronounced that it was insufficient. General opinion to-day has perhaps changed and there now appears to be a marked tendency to recognise the cogency of some of Wegener's general arguments and to pay greater attention to the possibilities which he propounded, because it is becoming increasingly clear that the question of the arrangement and of the stability of the primary structural units of the Earth's surface, during geological time, is one of the fundamental problems of geology, as well as of all the allied subjects of geophysics, geomorphology, geodesy, palaeoclimatology and palaeontology. This is the writer's opinion, for what it is worth, after studying the considerable literature on this important and fascinating subject which has accumulated in the course of the past 30 years, during part of a recent leave spent in “browsing” round the Schools and libraries of Cambridge University.     

OBSERVING WITH THE ZEISS AND WILD THEODOLITES

Abstract

RELATIVE to discussions on the Wild theodolite in the E.S.R. the following contribution considers some properties common to that instrument and the Zeiss II; perhaps to the Tavistock also, with which, however, the writer has had no experience on precise triangulation.     

THE CADASTRAL MAP

Abstract

In the Empire Survey Review for April 1934 (No. 12, pp. 382–4), the Editor has raised the question of the function of the cadastral map. As he remarks, the question is not simple, but it is easier to say what the function of the map is than to decide of what the map should consist in order that it may fulfil its function. Broadly speaking we may state that the function of the map is to record the boundaries of landed property in such a manner as not only to afford a pictorial representation but also to supply data for the identification of these boundaries on the ground when occasion requires. Apart from this the map should show the areas of properties, as this information may be required for taxation purposes.     

DEVIATION OF THE VERTICAL

Abstract

This paper endeavours to put forward good reasons, practical as well as academic, for needing a knowledge of the deviation of the vertical in all countries, and it then describes the instruments and methods used by the writer in his first season on the observation of a geoidal section through Great Britain. Unfortunately there has been no time as yet for the computation of the latter, so this cannot be a full report on it.     

LOCAL ATTRACTIONS AND THEIR INFLUENCE ON THE DETERMINATION OF HEIGHTS ABOVE MEAN SEA LEVEL

Abstract

The Mythical Spheroid.—The preceding article dealt with the fact that the spheroid of reference is a myth and that, even if it were not, we could not get hold of it at any given place. In order to apply corrections to observed quantities or, more generally, to operate upon them mathematically, we must make some assumption such as that of the spheroidal level surface. Probably a lot of harm has been done by attaching the notion of too concrete a thing to the spheroid. Disputes and misconceptions have arisen. People talk of“putting the spheroid down at a point” and imagine that the obedient thing is still at their feet when they get to another point, perhaps distant, in their system of triangulation or what not. Actually the spheroid may be disobedient not only as regards the direction of the vertical but also because it is above their heads or below their feet. What happens is that at each point afresh the computer treats the observations as if they were made there on the surface of a spheroid. In the same way, but travelling still farther along the road of hypothesis, he may treat observations for astronomical positions as if the compensation for visible elevations were uniformly distributed as a deficiency of density down to a depth of 122·2 kilometres. That was the depth which happened to give the smallest sum of squares of residuals in a certain restricted area, but nobody imagines that it corresponds with a physical reality, especially the ·2! It was a convenient mathematical instrument which, once the theory was to be given a trial, had to be fashioned out of some assumption or another. All this has little to do with geodetic levelling but is meant to try to banish the spheroid out of the reader's mind or at least to the back of his mind. In what follows we shall be compelled to make a certain amount of use of the family of spheroids but always with the above strictures in view.     

SOME NOTES ON COMPUTING

Abstract

The article that follows is a collection of notes on various aspects of computing, some of them rather elementary, which it is thought might be of interest to some surveyors. Most land surveyors spend quite a lot of time computing and many become expert at it, but even so they rarely reach the stage when they cannot usefully pick up hints from professional computers. It is therefore hoped that it may be useful to collect a number of points which the writer sometimes found helpful to surveyors during the time when he was doing arithmetic for a living.     

THE CONVERGENCE OF MERIDIANS

Abstract

Some publications that have dealt with the question of convergence of meridians seem, to the present writer, to be clouded with misconception, and these notes are intended to clarify some points of apparent obscurity. For instance, A. E. Young, in “Some Investigations in the Theory of Map Projections”, I920, devoted a short chapter to the subject, and appeared surprised to find that the convergence on the Transverse Mercator projection differs from the spheroidal convergence; the explanation which he advanced can be shown to be faulty. Captain G. T. McCaw, in E.S.R., v, 35, 285, derived an expression for the Transverse Mercator convergence which is equal to the spheroidal convergence, and described this as “a result which might be expected in an orthomorphic system”. Perhaps McCaw did not intend his remark to be so interpreted, but it seems to imply that the convergence on any orthomorphic projection should be equal to the spheroidal convergence, and it is easily demonstrated that this is not so. Also, in the second edition of “Survey Computations” there is given a formula for the convergence on the Cassini projection which is identical, as far as it goes, with that given for the Transverse Mercator, while the Cassini convergence as given by Young is actually the spheroidal convergence. Obviously, there is some confusion somewhere, and it is small wonder that Young prefaced his remarks with the admission that the subject had always presented some difficulty to him.     

THE ORTHOMORPHIC PROJECTION OF THE SPHEROID

Abstract

Chesterton did not, of course, intend this gibe to be taken literally. But the more we consider what he would doubtless have called the “Higher Geodetics”, the more we must conclude that there is some literal justification for it. Not only are straight lines straight. A sufficiently short part of a curved line may also be considered straight, provided that it is continuous (i.e. does not contain a sudden break or sharp corner), and provided we are not concerned with a measure of its curvature. Similarly a square mile or so on the curved surface of the conventionally spheroidal earth is to all intents and purposes flat. We shall achieve a considerable simplification, without any approximation, in the treatment of the present subject by getting back to these fundamental glimpses of the obvious, whether the formalists and conformalists accept them or not.     

RECONNAISSANCE MAPPING FROM TRIMETROGON AIR PHOTOGRAPHS IN THE ANGLO-EGYPTIAN SUDAN

Abstract

Trimetrogon air surveys have been used extensively in the United States and Canada but seem to have had less favour on this side of the Atlantic in spite of the fact that large areas over here were photographed in this way during the last war. The Sudan is a very large country, but not a rich one; and up to the present it has not been able to undertake its own aerial photography. The gift of a very large number of American trimetrogon photographs covering half the country was therefore very welcome, and it seemed worth our while to see what could be done with them in spite of the known disadvantages of this type of photography for anything but small scale mapping. The methods of planimetric mapping from these photographs developed in the United States were expressly designed to be as simple as possible and to be capable of division into a number of easy tasks which could be performed by any educated person after a short course of training. Nevertheless we felt that they could be simplified a great deal more if the problem was approached from a different angle, particularly in view of the very high standard of flying which was apparent in the photographs covering the Sudan.     

THE MODIFIED MERIDIAN DISTANCE IN THE CONICAL ORTHOMORPHIC PROJECTION

Abstract

Recently the writer of this article became interested in the conical orthomorphic projection and wanted to see a simple proof of the formula for the modified meridian distance for the projection on the sphere. Owing to the exigencies of the war, however, he has been separated from the bulk of his books, and, consequently, has had to evolve a proof for himself. Later, this proof was shown to a friend who told him that he had some memory of a mistake in the sign of the spheroidal term in m⁴given in “Survey Computations”, perhaps the first edition. Curiosity therefore suggested an attempt to verify this sign, which meant extending his work to the spheroid. This has now been done, with the result that the formula given in “Survey Computations”, up to the terms of the fourth order at any rate, is found correct after all.     

FURTHER NOTES ON RANGE-FINDING WITH A THEODOLITE

Abstract

Since writing the article which appeared in E.S.R., no. 36, p. 364, the writer has used this method to obtain thousands of spot heights with differences of elevation varying up to 7,500 feet in a single shot. The results of this experience and the modifications introduced in the method may be of some iilterest.     

EVALUATION OF THE COEFFICIENT OF TERRESTRIAL REFRACTION

Abstract

In a previous article on this subject (Empire Survey Review, January 1937) the writer sought to show that for trigonometrical observations of vertical angles made near noon in the Tropics the coefficient of refraction depends chiefly on height above ground level in the case of stations sited within a few hundred feet above the general level of the ground surface. Indeed, the computed values of the coefficient K show a definite and appreciable increase with “h”, the height of the observing station above ground level; it is usually assumed that K decreases with increase in height above the Mean-Sea-Level surface. From analysis of the results obtained by varying h but holding the heights above Mean Sea Level fixed the writer came to the conclusion that the variations in K could only be due to abnormal values of dt/dh and d²t/dh², “t” denoting the air temperature. Now it is generally recognized by meteorologists that abnormal lapse-rates of temperature do frequently occur in the lower air layers in the Tropics; but up to the present time no temperature soundings in Nigeria are available. Recently, however, the writer came across the results of the aerological soundings made by an expedition in East Africa during the year 1908. The results of many of the soundings were of no use for the purpose of this paper; many of the observations were not taken at or near noon, and in others counterlapses of temperature in the lower layers indicated that conditions were not normal. A set of observations taken at Mombasa between 10 and 11 a.m. were eventually chosen as offering an example of what might reasonably occur in the lower layers of the atmosphere.     

A MAP OF THE EMPIRE

Abstract

About sixteen years ago an apparent need of the time led the writer to consider the construction of a small-scale map of the British Empire. It was immediately obvious that such a sheet, prepared in a manner to reduce the errors of scale and bearing to figures approaching the minimum, would have to extend from the Yukon and across the Eastern Hemisphere to New Zealand_ Accordingly, it would involve an oblique projection, cutting the Equator at some angle to be investigated. Moreover, the depth of the map would necessarily be the least possible in order to conform with the desiderata above.     

AIR SURVEY AND THE PHOTOGRAPH

Abstract

As air-photographs are being more and more used for survey purposes, Empire surveyors who have not yet made a thorough study of aerial survey may be interested in a little elementary photographic geometry and its application to map-making. A map may be described as an orthogonal proj ection of the ground upon a horizontal plane, reduced to some convenient scale, and a photograph as a conical projection of the ground upon the focal plane of the camera. If the focal plane is horizontal at the instant of exposure and the ground being photographed is perfectly level, the two projections are exactly similar and the photograph is indeed a map. Unfortunately these conditions which are illustrated in fig. 1 are extremely rarely encountered, and photographs usually need correcting for various distortions.     

THE UGANDA MAlLO SYSTEM

Abstract

The survey of “mailos” or native estates in the Kingdom of Buganda has taken a prominent place in the annual programme of the Survey Department of Uganda for over 30 years past. The survey, which has covered some 17,000 square miles and is now practically complete, has some unusual features, and although it has no claims to refinement or to great precision, a short account of its history and workings may be of general interest. The system of land settlement introduced by Sir Harry Johnston has already been described in the Empire Survey Review (“The Surveyor and the Politician”, by H. B. Thomas, vol. ii, p. 28).