TACHEOMETRIC SURVEYING AND THE USE OF THE DIFFERENTIAL SLIDE RULE

Abstract

In the E.S.R. No. 17 of July 1935, page 138, there appeared an article by Prof. F. A. Redmond on “The use of Even Angles in Stadia Surveying”. Since I have given this method a six-months' test in the field, using Prof. Redmond's “Tacheometric Tables” for the reduction of the measurements, the conclusions reached may be of some interest.     

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.     

GEODETIC BEACONS

Abstract

Mr. Clendinning's article on “Signal Lamps” (E.S.R., vol. ii, pp. 15–18) raises a point of major importance in geodetic triangulation. I entirely agree with him that the sole use of heliographs—heliostats to the purist—is in most parts of the world out of date. I also think, and indeed am prepared to state categorically, that the use of acetylene lamps is out of date and was out of date many years ago. The Americans, who are always worth listening to on the economics of surveys, would not otherwise have replaced all their acetylene gear by electric beacons. The answer, in my experience, and for reasons which I shall endeavour to make clear, is generally, but not necessarily always, to provide both helio and electric lighting; but first I should like to summarize the conditions in which luminous signals should be used at all.     

FIGURES OF REFERENCE FOR THE EARTH

Abstract

1. The object of this note is to clear up what I believe to be some misconceptions regarding the use of a reference system by a surveyor of the earth's surface. In his article “An Aspect of Attraction”, E.S.R., No. 7, pp. 24–8, Major M. Hotine expressed doubts as to the validity of the process usually followed. I may say at once that I consider these doubts are unfounded.     

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.     

A GREAT BRITISH GEODESIST ALEXANDER Ross CLARKE (1828–1914)

Abstract

That man is to be envied who can devote many of the best years of his life to the study of a special branch of science and make some advances in it. Such a man will usually receive recognition of the value of his labours from his fellows in the world of science, and this was certainly the case with Colonel Clarke. The excellence of his many years' work on geodetical subjects, such as thereduction of observations, formulre for the spheroid, figures of the earth, standards of length, and similar matters, was fully appreciated by scientific men during his lifetime, in this country as well as abroad. Curiously enough, his name does not appear in the “Dictionary of National Biography”, though he is, perhaps, the best known of British geodesists. A paragraph is devoted to him in recent issues of the “Encyclopredia Britannica”, but this paragraph is, in one respect, inaccurate. One may say that geodesy makes little appeal to the ordinary citizen, who usually would not know what it is all about.     

SURVEYING INSTRUCTION AT THE UNIVERSITY

Abstract

The title and content of Mr. Ray's contribution to the July number of the E.S.R. entitled “Surveying Instruction at the University” is in the present writer's opinion, somewhat misleading. Obviously he is only referring to the teaching of surveying to civil engineering undergraduates and to students of such other branches of engineering as may be catered for, and does not mention the quite extensive surveying courses associated with the degrees in Geography and Mining at certain Universities. The gist of the article is confusing in that the author concludes by stating that he is “making an honest attempt to assess the merits of the surveying instruction given in the Universities of this country”, which is an admittedly worthy quest but scarcely compatible with some earlier statements such as “the long overdue overhaul of our University courses”, a statement which seems a little premature for one who is still in the stage of “honestly assessing” what is going on.     

BATTLE SONG OF THE FAR-FLUNG SURVEYORS

Abstract

Air.—Any good Guest Night tune which happens to fit to a first order, and remains more or less in tune after subsequent orders. “Coming down the Mountain” and “Kabul River” would do.     

THE NOMENCLATURE OF MAP PROJECTIONS

Abstract

Map Projections.—A matter that should have been mentioned in the original article under this title (E.S.R., vii, 51, 190) is the definition of a map projection. In the list of carefully worded “Definitions of Terms used in Surveying and Mapping” prepared by the American Society of Photogrammetry (Photogrammetrie Engineering, vol. 8,1942, pp. 247–283), a map projection is defined as “a systematic drawing of lines on a plane surface to represent the parallels of latitude and the meridians of longitude of the earth or a section of the earth”, and most other published works in which a definition appears employ a somewhat similar wording. This, however, is an unnecessary limitation of the term. Many projections are (and all projections can be) plotted from rectangular grid co-ordinates, and meridians and parallels need not be drawn at all; but a map is still on a projection even when a graticule is not shown. Objection could be raised also to the limitation to “plane surface”, since we may speak of the projection of the spheroid upon a sphere, or of the sphere upon a hemisphere. Hence, it is suggested that “any systematic method of representing the whole or a part of the curved surface of the Earth upon another (usually plane) surface” is an adequate definition of a map projection.     

THE STRAIGHT POSITION LINE

Abstract

The article “Notes on the Position Line” by B. Chiat (E.S.R., xiii, 97, 137) is very informative in the conclusions reached regarding the validity of drawing the position line straight, but it seems, to me at least, that the discussion involving the effects of the earth's non-sphericity is an academic labouring of a difficulty which, in fact, is non-existent.     

THE STEREOGRAPHIC SOLUTION OF THE SPHERICAL TRIANGLE

Abstract

In the course of his stimulating and suggestive paper in your recent issue, No. ro, pp. 226–38, Mr. A. J. Potter writes on p. 233 “but there is no simple construction by which X can then be found”, and again on p. 237 “a direct construction, if there be such”. This cheerful challenge invites the construction of a circle centred on a given line, passing through a given point thereon, and touching a given circle, and I have found the lure of Mr. Potter's gauntlet as irresistible as its recovery has proved delicate. In order to shoulder responsibility and by no means to claim highly improbable originality, let me confess that the problem is new to me and the two constructions I offer are my own; I venture to hope that Mr. Potter may consider one or other of them not unworthy of his epithet “simple”, though I freely admit the aptitude of his empiric procedure to its purpose. The proofs are not long, but for fear of overshooting my welcome I offer them to anyone for the asking; and for the same reason my diagrams are small and therefore mere.     

ERRORS OF THE PARALLEL-PLATE MICROMETER

Abstract

The last issue (No. 26) of the Review contained an article on “Observing with the Zeiss and Wild Theodolites”, making reference among other matters to the errors of the parallel-plate micrometer. The statement was made that the error was due to the difference in travel between the two plates. This is not strictly correct but could not be better expressed without additional explanation, out of place in an already overlong article.     

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.     

THE TWO METRES: THE STORY OF AN AFRICAN FOOT

Abstract

Introductory.—From time to time the question of the relation between the metre and the foot is raised, most frequently perhaps from Africa. Had there been no more than a single metre to consider the question would no doubt arise but seldom: the most recent authoritative comparsion would be generally accepted. But actually it is the existence of two metres—the “ legal” and the “international”—which complicates the question, so much indeed that there is no metrological factor which has influenced survey, British and foreign, more than the relation between these two metres. The question was discussed in this Review (I, 6, 277, 1932), but memories grow shorter, attention is more diffused, and besides there is required a more explicit statement of the situation as it affects British surveyors, especially in Africa, whence the question has been raised anew. To illuminate it, unfortunately the need recurs to repeat some well-known facts.     

THE CONFORMAL TRANSFORMATION OF A NETWORK OF TRIANGULATION

Abstract

In a recent issue of this Review, an example is given of the conformal transformation of a network of triangulation using Newton's interpolation formula with divided differences. While the application of the method appears to be new, attention should be drawn to the fact that Kruger employed Lagrange's interpolation formula in a discussion and extension of the Schols method in a paper which was published in the Zeitschrift für Vermessungswesen in 1896. A reference to this paper was given at the end of the paper, “Adjustment of the Secondary Triangulation of South Africa”, published in a previous issue of the E.S.R. (iv, 30, 480).     

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.     

THE ADJUSTMENT OF A FIRST-ORDER LEVELLING NETWORK

Abstract

In the E.S.R. January and April numbers of 1955, Vol. xiii, Nos. 95 and 96, Mr. Hsuan-Loh Su described the “Adjustment of a Level Net by Successive Approximations and by Electrical Analogy”. It does not seem to be as generally known as it should be that the rigid least square solution can be greatly simplified by utilizing the electrical analogy and solving by Kirchhoff's method. The method as detailed below has been in use for over 40 years.     

RELAXATION AND THE CO-ORDINATE METHOD OF TRIANGULATION ADJUSTMENT

Abstract

Few, most certainly, will dispute the value of Mr Black's paper describing a method of “Systematic Relaxation”, which appeared in a previous number of this Review. At the same time, however, it seems to the writer to be only fair to readers to point out that the application of the method to triangulation adjustment is really a treatment, from a slightly different aspect, of methods that have long been established.     

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.