A report that the old-established division of the right angle into 90 degrees is to be superseded in Germany by a centesimal division into grades, seems to have had considerable publicity in the press. In spite of the startling headline, which is reproduced above, the news paragraphs are rather restrained in that they state that the edict is to apply immediately only to the teaching of the new system; we are led to believe, however, that in a few years the use of the centesimal division of the quadrant will be obligatory. In view of the far-reaching effects of such a change it is timely to examine both the authority of the report and the possibility of such a geometrical revolution in Great Britain.

Enquiry from leading German mathematicians and astronomers has revealed that the reports have been greatly exaggerated; it is impossible to believe that any decree involving such a great revision of mathematical and astronomical tables, instruments and records, to say nothing of the confusion bound to arise in minds inured to the old system, could have been promulgated without the knowledge of the leading authorities in mathematics and astronomy. What importance and what weight are we then to attach to such a report? It would seem probable that an attempt is being made to introduce the centesimal system gradually, (but not by degrees!), in the first place by teaching the new system in schools, by using it wherever possible for theoretical calculations and possibly by extending it to geodetic work. That is exactly the extent to which this system has progressed in France (the home of the metric system) after 150 years. As pointed out by Professor Borel in his summary of the position in France, a further extension to cover the two main uses in astronomy and navigation demands international agreement. There must only be a small chance of this, and even if the principle were approved, how many generations would be necessary before the new system could be safely brought into universal use?

Note. The John Todd referred to here is in our Archive as Jack Todd.

In 1942, Prof J A Carroll, shortly after joining the headquarters staff of the Director of Scientific Research in the Admiralty as assistant director, saw that scientific effort in the Admiralty establishments could not be fully employed in its proper field because scientific men were having to undertake their own computational work. Often, for lack of training in computational mathematics, they spent even more time on this than it needed intrinsically and, more serious still, for lack of internal or external facilities, some investigations requiring extensive computations were foregone. An investigation of the computational needs of establishments by Mr D H Sadler, superintendent of H M Nautical Almanac Office, was accordingly arranged. Earlier, the Nautical Almanac Office had been doing large-scale computational work for other Government departments, notably the Ministries of Aircraft Production and Supply, in addition to its navigational work for the Air Ministry; proposals for an inter-departmental computing organisation which had been made earlier by the Office were overtaken by the joint proposals, referred to later. Following Mr Sadler's report that a considerable amount of computational work could with advantage be centralised, a mathematical section was set up in the Department of Scientific Research and Experiment to deal with this work and with the more difficult mathematical problems which were known to arise in naval establishments and which were being either set aside or tackled by totally inadequate methods. This section became known as the Admiralty Computing Service. Mr John Todd was responsible for organisation and supervision of this Service. Staff were attached to the Nautical Almanac Office to carry out the actual computations under the direction of Mr Sadler, and arrangements were made whereby scientific workers in the universities and elsewhere could be employed as consultants.

Note. The John Todd referred to here is in our Archive as Jack Todd.

In 1942, in order to use more efficiently the scientific staff available in the Admiralty, the Director of Scientific Research set up, within the branch directed by Dr J A Carroll, an Admiralty Computing Service to centralise, where possible. the computational and mathematical work arising in Admiralty Experimental Establishments.

Mr John Todd undertook the organisation and supervision of the Service. By agreement with the Astronomer Royal additional staff were attached to H M Nautical Almanac Office to carry out the computational work under the direction of the Superintendent, Mr D H Sadler. In addition, arrangements were made to permit the employment of experts from the Universities and elsewhere as consultants.

The work undertaken by Admiralty Computing Service was in general of one of two classes: heavy computation, or difficult mathematics. Altogether more than one hundred separate investigations were carried out ranging from projects involving several thousand hours' computing to small problems for which a solution could be obtained in a few hours. In addition, a considerable amount of advisory work has been undertaken, usually informally. For instance the Five-figure Logarithm Tables were designed by Admiralty Computing Service for the Ministry of Supply, as one item in a comprehensive program for providing the optical industry with the tables they required. For various reasons, mainly owing to the increased use being made of machines and to the availability of the U.S. reprint of Peters' seven-figure table of natural trigonometric functions, other tables were never published.

Shortly after the formation of Admiralty Computing Service it became apparent that research work in Admiralty (and other) Establishments would be greatly facilitated if their members were informed in certain mathematical and computational techniques not usually covered in undergraduate courses, and of which no adequate account was available in easily accessible literature. Accordingly the preparation of a series of monographs of an expository nature was begun.

It was soon realised that while centralisation within a Department was an improvement, nothing less than centralisation on a national scale could be really efficient. At the end of 1943, an approach was therefore made to Sir Edward V Appleton, Secretary of the Department of Scientific and Industrial Research, asking for consideration of the formation of a National Mathematical Laboratory. Discussions, in which the experience gained by Admiralty Computing Service played an important part, have now resulted in the formation of a Mathematics Division of the National Physical Laboratory. Staff have been released from Admiralty Computing Service to form a nucleus for the computational sub-division of the new organisation. It is anticipated that the computational needs of the Admiralty will be met by outside organisations and the mathematical needs by an even larger use of the service of consultants, working under the general direction of the Director of Physical Research, Admiralty.

The title of this lecture was chosen over a year ago before I had formed a clear picture of its scope. It is, however, adequately descriptive of my purpose which is to develop consistent general principles, as opposed to detailed methods, which can serve as a guide to the individual computer - primarily one working alone with an ordinary calculating machine. I shall endeavour to illustrate the principles by actual examples, but it will be outside my present compass to concentrate upon those methods which rely upon brilliance of technique rather than on soundness of principle.

It is clearly desirable to say precisely what 'computation' embraces, and I therefore put forward the following definitions of 'calculation' and 'computation' - in each case it being understood that the word is prefixed by 'numerical':

Calculation is the application of the four rules of arithmetic to the numerical evaluation of an algebraic expression.

Some apology, or at least an adequate excuse, is needed for resurrecting a theoretical treatment of the effect of Coriolis acceleration on observations of altitude made with a bubble sextant. Such an excuse is provided by the recent publication by Dr J J Green of an article suggesting that the correction table (Z-correction) given in both the British and American Air Almanacs (and in many other Air Almanacs) is incomplete. In a reasoned letter to the Editor of Navigation, Dr G M Clemence, Director of the American Nautical Almanac Office, has given a simple and straightforward explanation of the two separate and distinct causes for the deviation of the zenith as indicated by the bubble of a bubble sextant; and he has further justified the present practice adopted in the almanacs. Considerable interest has, however, been aroused and it seems opportune to give a previously unpublished general derivation of the theoretical correction, together with a brief discussion of the difficulties of practical application.

Milne's formula for approximate quadrature is used as the basis of a method for the solution of first-order differential equations on the National machine. The method, which is illustrated by a numerical example, enables the machine to form the required dependent variable without the necessity for conversion from a sum to an integral.

When, last year, you did me the honour of electing me your President you gave me the duty, and the privilege, of composing and delivering a Presidential Address. It does not lie within my abilities to make a grand survey of the present state of any great and important navigational development - such as might fittingly form the central theme of such an Address, and which has, in fact, been so admirably treated by my predecessors in this office. I have, however, been privileged to serve on the inner councils of the Institute since the earliest days of its conception, and feel that I may be in a position to take the Institute of Navigation itself as a central theme. I do so at this time particularly because the by-laws of the Institute quite rightly operate to ensure that no one person shall remain on the Council indefinitely and I must be the last to have such continuous service since the foundation. But there is a more cogent reason: the Institute was founded in the enthusiasm of the immediate post-war application of war-time navigational methods to civil use, and its precise role was deliberately left vague. It will be my object this afternoon to examine the extent to which the Institute is fulfilling its initial purpose and, on the basis of seven years' experience, how it can best play its part in the future.

Of the many factors affecting the collision problem, the mathematics of relative motion is the one most susceptible to formal analysis. Moreover, this 'collision geometry' must form the basis of any investigation into the instrumental, operational and human factors involved. It is therefore surprising that no complete description of the (mathematically) simple relationships is readily available. The object of this paper is to present such a connected account of the mathematics of relative motion, in so far as it appears relevant to the collision problem and within the limitations of time and space available.

Most people think that close artificial satellites of the earth fall directly within the field of responsibility of H M Nautical Almanac Office. This is not so. The Office, and the corresponding ephemeris offices in other countries, are concerned primarily with long-term, high-precision theories and predictions of heavenly bodies; comets and meteors are excluded from our field and are dealt with competently by others, mainly by amateurs. Our professional interest will be aroused when an artificial satellite is launched into an orbit with perigee so high above the earth's surface that atmospheric drag is very small; such an object, with a life of ten years or more, will be of definite astronomical importance. The ephemeris offices will then produce, and publish, accurate ephemerides based on a combination of theory and observation to assist in the comparison of theory with observation. It is conceivable that such satellites will provide rapid and accurate methods for the determination of the second of ephemeris time, now accepted as the fundamental standard of time. Navigational applications must also be considered.

For the two satellites so far (29 November) launched by U.S.S.R., the Office has been compelled, through limitations of staff and equipment, to confine its work to

(a) a prediction service, and

(b) the collection and copying of observations and their transmission (in due course) to I G Y World Data Centres, and possibly directly to other institutes.

Captain Majendie would restrict the use of the word blunders to those aberrations of human behaviour which cannot be attributed to any avoidable cause. For my purpose this definition is unduly narrow and I use the concept in a slightly wider sense, namely: 'errors, or aberrations, without apparent cause'. An understanding of the underlying causes of blunders is essential to their proper study, and to the reduction of blunder rates. Assuming that these causes are factors of human behaviour, there should be some connection between the occurrence and causes of blunders in different fields of human activity. In the hope that this may be so, I summarise the conclusions drawn from investigating blunders over a period of 25 years with a staff of 20-30 engaged on numerical computation and proof-reading.

A meeting of the Technical Committee was held on 9 January in London, at a session which was open to Members of the Institute and to various other people by invitation. From the Chair Wing Commander E W Anderson explained that the object of this kind of meeting was to enable the Institute to consider more or less informally matters which were in dispute and to give all Members an opportunity of attending these discussions. The subject under consideration had been discussed, sometimes with vehemence, at meetings, in private correspondence, and in The Journal but agreement among the contestants seemed no nearer. An exchange of views at a technical session might be fruitful. In this account of the discussion, written contributions not presented at the meeting have been included.

Note. Text of a talk given at the Ordinary Meeting on 9 December 1966.

On the ninth day of February 1765, Nevil Maskelyne, fifth Astronomer Royal, presented to the Board of Longitude his suggestion for the calculation and publication of a 'Nautical Ephemeris', designed to make possible the determination of longitude at sea by means of the method of lunar distances. The Commissioners of Longitude resolved 'that Application should be made to Parliament ... for power to give a Reward to persons to compile a Nautical Ephemeris and for Authority to print the same'. The application was duly approved and the first edition of The Nautical Almanac and Astronomical Ephemeris was prepared for the year 1767 and published in 1766. At the same time Maskelyne also published his Tables Requisite to be Used with the Astronomical and Nautical Ephemeris to serve as a handbook for use with the Almanac.

Although the motivation behind the publication was almost entirely navigational, the Almanac (as its full title indicates) also contained what are now known as 'the fundamental ephemerides of the Sun, Moon, planets and stars' to the highest precision. At that time, as will be seen, they were barely adequate for the purpose of navigation; but, with the advancing progress of astronomical observation and theory, the increasing precision outstripped that required for navigation. Even so it was not until 1914 that the two distinct functions of The Nautical Almanac were recognised by the publication of separate almanacs, one for navigators and one for astronomers. The position was ultimately clarified only in 1960 when the titles of the two publications were changed to the appropriate parts of the original title: The Nautical Almanac is specially designed for the use of marine navigators; and The Astronomical Ephemeris has no navigational use. There are now also The Air Almanac for the use of air navigators, and The Star Almanac for the use of land surveyors - a total of some 1700 pages annually as compared with the original, much smaller, 165 pages. It is better to make no comment on matters of productivity and efficiency, bearing in mind that astronomy now provides a very minor aid to navigation!

Note. Delivered at the Anniversary Meeting on 9 February 1968.

The duty and responsibility of preparing a Presidential Address is a heavy one, but it has its compensations. It provides a unique opportunity to speak to the assembled Fellows of the Society, without fear of interruption or opposition, and without examination by discussion. There is a good chance that one can get away with opinions that one might not otherwise care to express in public. Several Presidents have taken full advantage of this 'presidential licence' in the past, but I propose only to do so in a mild and gentle way.

Those of us who are old-fashioned enough to work in the field of positional astronomy have recently been assailed by the physicists who, having developed atomic clocks of incredible accuracy, now consider that the natural time-scales of astronomy can be replaced by the natural frequencies of atomic or molecular transitions. I hope to convince you that this is not so.

We have also been assailed by the politicians, to whom centuries of scientific tradition and usage are apparently of little account compared to the fleeting and meaningless results of a newspaper competition; I refer, of course, to the name now being proposed for permanent Summer Time. But I shall not abuse this platform by deigning to say any more on this subject than how fortunate we are that we were able to commence business at this Anniversary Meeting of 1968 at 'half-past four o'clock in the afternoon' Greenwich Mean Time, which is currently also British Standard Time.

As far as I can discover this is the first occasion on which a President of the Society has chosen the subject of 'time' for his Address. I do so now for two reasons: firstly, to try to explain some of the mystique that has arisen in connection with Ephemeris Time, and, secondly, to clarify the respective roles in astronomy of Ephemeris Time and Atomic Clock Time.

Before doing so I would like to review, briefly, the historical relationship between astronomical determinations of time and 'mechanical' time-keepers or clocks; in doing this I shall try to establish some concepts, definitions and terminologies that may be useful in understanding the main objectives.

My first reaction on being invited by the Editor to contribute a 'twenty-first birthday' article on the above subject for the Journal was that there is little new to say. However, it is now 13 years since I prepared the presidential address on 'The Place of Astronomy in Navigation' (this Journal, 9, 1, 1956) and, apart from any other developments, space navigation has since become established as an operational science as well as a theoretical discipline. There have, in fact, been many developments and much progress in these 21 years; and, although these appear in the Journal and elsewhere mainly as gradual changes, a comparison over the whole interval reveals drastic changes, particularly in outlook and appreciation.

The Royal Observatory was founded in 1675 specifically for "… the rectifying the tables of the motions of the heavens, and the places of the fixed stars, so as to find out the so much desired longitude of places for the perfecting the art of navigation." Ninety-one years later Nevil Maskelyne, the fifth Astronomer Royal, was able to compile The Nautical Almanac and Astronomical Ephemeris for the year 1767 which made possible the determination of longitude at sea to an acceptable precision.

The key to the successful solution of this problem is the method of lunar distances, and this method is discussed both historically and scientifically.
The objects of this paper are: to examine the contribution to this achievement made by the Royal Observatory and its Astronomers Royal; and to speculate on the reasons for, and the effect of, the adoption of a nominal precision of 13 (quite unnecessary accuracy) in the tabulations of lunar distances and in the associated calculations.

All the main historical events and developments are well-known, and the historical detail has been curtailed accordingly.

The object of this expository article is to review certain aspects of the definitions of, and terminology for, mean solar time on the meridian of Greenwich, and to explain the effects on astronomy of adopting the recommendations recently endorsed by the General Assembly of the International Astronomical Union.

The object of this paper is to describe, in principle, a method of reducing astronomical sights, of plotting the resulting position lines and of calculating a position from two or more sights. The method, which is thought to be new, requires neither a calculator nor extensive tables and appears to possess some advantages over more conventional methods. After a general description, and illustrations of two variations, the method is analysed in sufficient detail to indicate that its practical realisation, to meet any specific requirement, is a straightforward matter of detailed design.

No attempt is here made to translate the principles of the method into a form for practical use. The paper is mainly addressed to those interested in the theory of astronomical navigation; the description, together with the analysis and the illustrations, is available for use as a basis for a practical method of sight reduction - should anyone consider that there is a requirement! The treatment is intended to be rigorous, though much elementary mathematical detail has been omitted; in a practical application some care may be required near the singularities.