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.     

THE TRANSVERSE MERCATOR PROJECTION OF THE SPHEROID

Abstract

In January 1940, in a paper entitled “The Transverse Mercator Projection: A Critical Examination” (E.S.R., v, 35, 285), the late Captain G. T. McCaw obtained expressions for the co-ordinates of a point on the Transverse Mercator projection of the spheroid which appeared to cast suspicion on the results originally derived by Gauss. McCaw considered, in fact, that his expressions gave the true measures of the co-ordinates, and that the Gauss method contained some invalidity. He requested readers to report any flaw that might be discovered in his work, but apparently no such flaw had been detected at the time of his death. It can be shown, however, that the invalidities are in McCaw's methods, and there seems no reason for doubting the results derived by the Gauss method.     

REPORTS. THE YEAR 1934 IN CEYLON

Abstract

The Ceylon Survey Department is gradually returning to its normal activities after the period of depression. The programme of work is again being increased as the demand for survey work grows with returning prosperity in the Colony. The Survey Department Training Classes have been reopened for training new recruits to fill the vacancies in the upper and lower grades which have been left unfilled during the retrenchments. It takes a year, however, to train a subordinate office: and two years to train a probationary Assistant Superintendent, Including one year at Cambridge. Recruiting is consequently rather. a slow process, and it will be some years before the Department is again at full strength, as there are now 27 vacancies in the Field Staff and 34 in the Office Staff to be filled, out of a total of 705.     

THE PLACE OF MOUNT EVEREST IN HISTORY

Abstract

The Himalaya are known as the highest mountains of the earth; but they have another title to fame not generally recognized in that they have been the home of civilized man for a longer period than any other mountain range. When the monoliths of Stonehenge were being raised in England and when the Pyramids were being built in Egypt, the glaciers of the Himalaya were being explored. The geographical names which are entered upon modern maps of the Central Himalaya are very ancient memorials of civilization. The existence of a mountain nomenclature, Aryan in the west and centre, Mongolian in the east, has given rise to linguistic problems not less difficult than the topographical problems of the snows. In 1927 the volumes of the Linguistic Survey of India were published; they had been written by Sir George Grierson, an officer of the Indian Civil Service, and they dealt with 723 different languages and dialects. In the Himalaya there are now in use among the peoples seventy different languages, all of which (with one aboriginal exception) belong either to the Aryan or Mongolian family. Languages are historical monuments, and from them Sir George Grierson has been able to decipher the outlines of history.     

THE RADIAL LINE ASSUMPTION

Abstract

It is very many years since the Radial Line method was first used in America and England, and, so far from going out of favour, as was predicted by some European surveyors, it is now thriving in the Slotted Template form. The history of the method is an example of the Anglo-Saxon genius for persevering in a practical compromise. Had we been tempted by the voices of the photogrammetric Rhine Maidens to adopt theoretically exact methods to solve all problems, virtually no new mapping would have been carried out during World War II and many thousands of square miles of sparsely populated territory in the U.S.A. and British Commonwealth would still remain unmapped. A much greater mapped area is associated directly or indirectly with such names as Bagley and Hotine than with others more familiar at international conferences.     

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 NAMING OF THE STARS

Abstract

We have seen that a few constellation-names used by us to-day make their appearance as far back as the Homeric poems, while the star-names Sirius and Arcturus are found in the works of Hesiod. We saw no reason to suppose that these early poets knew no names for other groups. They mentioned only those conspicuous heavenly objects that might occur in any general literature, as for example in the Old Testament books of Job and Amos. There are sporadic references to constellations in the extant works of the great Greek tragedians of the fifth century B.C., but these give us few additions to the Homeric list'. It is not till we come to the special literature of the subject that constellation-names appear in any quantity. The first specialist whose name stands out is the mathematician Eudoxus of Cnidus, a disciple of Plato, who lived in the first half of the fourth century B.C. He wrote a prose work entitled the “Phaenomena”, which, though unfortunately lost, was rendered into verse by the poet Aratus about 275 B.C.     

THE WASHINGTON (1939) CONFERENCE OF THE UNION OF GEODESY AND GEOPHYSICS

Abstract

For the first time in the Western Hemisphere the Union met in conference in Washington between the dates 4th and 15th September 1939. Though rumours of indefinite postponement were current in the Press, there was never any intention that the Conference should not be held according to plan, which action was supported throughout by the State Department, without whose effective co-operation the event could not have succeeded. The fears of world-wide cataclysm, too soon turned into grim reality, no doubt influenced the attendance of European delegates; but the influx of representatives of the United States and Canada was correspondingly greater, and contributed to the success of the assemblies. In a sense the Conference gained by the reduction of members, for more time was available for the cultivation of former acquaintanceships; moreover, in the fraternizing of the European delegates with American scientists an opportunity was given to reveal the individuality behind well-known names, which helps so much towards mutual understanding.     

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.     

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.     

REFORM OF THE CALENDAR

Abstract

The majority of readers are doubtless aware of the masterly summary of the “History of the Calendar,” written for the Nautical Almanac for 193I (pp. 734–747) by Dr. J. K. Fotheringham. Most are probably also aware that the question of Calendar Reform has been considered by the League of Nations. At the Conference on Communications and Transit of 1931, October 19, the League adopted a resolution recommending a fixed Easter, but declared that “the present time is not favourable … for considering … a reform of the Gregorian calendar.” For information on the various measures of reform proposed at Geneva the works noted below may be consulted. In the meantime, pending the coming of reform—for come it will—readers may desire to.have a summary history of the question, with a statement of a solution which is of somewhat the same nature as others which have been proposed.     

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.     

AZIMUTH COMPUTATION IN THE OLD PRINCIPAL TRIANGULATION OF THE BRITISH ISLES

Abstract

The old Principal Triangulation of Great Britain and Ireland is a classic, both in execution and publication. The names of many of those who planned it, made it possible, carried it out or computed it have become legendary in the Ordnance Survey. Such are Roy, Dalby, Ramsden, Mudge, Colby, Drummond, Steele, O'Farrell and, greatest of them all, Clarke.     

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.     

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.     

THE ORDNANCE SURVEY

Abstract

If, one and all in our several ways, we are absorbed in helping to win the war, there are moments when it is possible to turn to our earlier loves, if but to relieve the strain. Not one of our hobbies, not a single insti tu tion that holds our affections and our memories, can emerge from this struggle just as it was. New thoughts and new methods are in the air, new service will be required, and, if principles remain largely unaltered, they, and the lessons we have learned, will be in peril, for continuity and knowledge have been rudely interrupted. It may do no harm to restate, however inadequately, some of those lessons—political, administrative and technical.     

THE LEVELLING OF A THEODOLITE

Abstract

A Fully equipped theodolite is provided with plate levels, an alidade level, and a striding level. An instrument not so equipped has no title to be considered a “Universal Instrument”, that is to say, an instrument designed for every kind of both terrestrial and celestial measurement. Without a striding level, for example, nothing beyond relatively rough astronomical measures can be expected in general. Modern instruments, capable of giving considerable refinement in terrestrial measures, are frequently not furnished with a striding level; and it is sometimes assumed, with the tacit approval of the makers, that such instruments are equally capable of giving refined astronomical results. On the older type of instrument a striding level—rarely not supplied—could have been, and sometimes was, extemporized; it seems as if ignorance of astronomy of position has led, at least in part, to the construction of theodolites in such a manner as actually to render such extemporization difficult.