MATHEMATICS AT OSBORNE AND
DARTMOUTH.

INTRODUCTORY.

A brief account of some of the conditions and requirements at the Osborne and Dartmouth Royal Naval Colleges may be of service as a preface to Mr. Mercer's more detailed discussion of the scope of the work done there.

It must at the outset be understood that the account is entirely unofficial and, so far as it deals with matters of opinion, it expresses nothing more than the writer's individual views. It should also be borne in mind that the syllabuses and methods pursued in the Naval Colleges are elastic and are liable to modification as experience may suggest or as the changing requirements of the service may render desirable.

The founding of Osborne College was associated with a lowering of the age at which naval cadets enter the service. They now join when they are barely 13. This lowering of the age of entry made it possible to arrange a consistent curriculum from 13 to 17, which should give the emphasis to the humanities essential in the education of such young boys, while by omitting Latin and Greek a greater proportion of time than is usual at public schools could be devoted to Science, and provision could be made for a thorough practical training in engineering workshops. This last could not be obtained in the schools which formed the recruiting area for naval cadets, partly because of the costly nature of the requisite plant, and partly through fashion and tradition.

Every executive naval officer has now, in the performance of his ordinary duties, to make use of complex mechanical and electrical appliances. He cannot rely on calling in an expert to deal with them, but must himself be in some sense an engineer; so his early training should be planned with due recognition of this fact.

At Osborne and Dartmouth roughly one-half of the working time is devoted to Science, Engineering, and Mathematics; of this time from one third to a half is spent in the workshops. The time devoted each week to Mathematics during the two years at Osborne is about 6 teaching hours and 2 hours of preparation. At Dartmouth during the first year it is about 5 teaching hours and 24 hours preparation, and during the second year 6 teaching hours and 3 hours preparation; but this includes also the time for navigation, which occupies about 3 hours weekly during the second year. After cadets have passed out of Dartmouth they spend 8 months in a special cruiser,

where a still more practical turn is given to the training. The abler boys here carry on their study of pure mathematics, in such time as they can gain through their greater rapidity in mastering other subjects of instruction. The time which can be found in this way cannot be stated exactly.

It will be seen that the aggregate amount of time available for Mathematics is no greater at these Colleges than on the Modern Side of most schools. This indicates clearly a deepseated change which has been made in the training of naval officers; speaking broadly that training was formerly mathematical, while now it is much less specialised, but has a strong leaning towards mechanics.

A considerable knowledge of the theory and practice of navigation and pilotage is, and has been for many years, an essential for every naval officer; formerly this was the only part of his professional training which could conveniently be begun on shore. In former times, when this shore training ended at 14, it was difficult to provide a sufficient mathematical equipment on which to base it, and educational balance had to be sacrificed for the purpose. As the age of entry rose, since no other claims asserted themselves strongly, the emphasis on Mathematics persisted. In the latter days of the "Britannia," cadets spent at least 14 teaching hours and 5 hours of preparation weekly on Pure Mathematics and Navigation, and an exceptional amount of time had to be given to Mathematics in the education_leading up to the competitive examination for entrance. It is now impossible to assign so preponderant a position to Mathematics; and since the demand for efficiency in Navigation is no less imperative now than formerly, it is necessary to devote very great attention to the development of the course in order that it should be possible to cover the necessary ground in the decreased time available.

Some changes in the practice of Navigation facilitate this; increased reliance on chronometers and the consequent abandonment of lunars as a matter of routine, and the introduction of the new Navigation into the British Navy, have considerably reduced the mathematical skill required of the ordinary navigator. Great economy of time results from recent changes in the methods of teaching Mathematics, and still more can be gained by eliminating large portions of the subject which have become traditional, but whose chief justification depends on their utility in defeating an examiner. They have come into existence mainly through the ingenuity of earlier examiners and writers of text-books, not through the demands of the man who uses Mathematics as a tool or who wishes to carry it on to higher levels for its own sake. There seems, too, to be a limit to the amount of time. which can be spent with profit on a restricted range of elementary mathematics; after this limit is exceeded the work becomes deadening and mechanical. In the particular subject of navigation this is not a serious drawback, as the ordinary processes

must be practised so often as to become almost automatic. Finally it must be recognised that Mathematics enters very largely into the subjects which have partly replaced it, and that such applications not only provide practice but also enormously vivify a boy's conception of what he learns in a mathematical class-room.

A naval officer requires Mathematics for use in subjects other than Navigation, no less essential to him.

Throughout the four years and a half of training at Osborne and Dartmouth and in the training cruiser, there is a continual call for mathematical knowledge in his work in engineering and science which forms the basis of subsequent study in the separate branches of gunnery, torpedo and engineering. (It may be worth while to explain that the second of these includes electrical engineering.)

A certain proportion, probably less than half of the cadets, will not need to pursue the study of higher mathematics at later stages of their career; they will form the body of "general service officers" from whom there is not required a greater knowledge of navigation, engineering, science, etc., than can be based on the amount of Mathematics acquired before leaving Dartmouth.

But foundations have also to be laid for the future study of more advanced Mathematics in the case of those capable of it. Of the remaining cadets who will be needed for specialist officers, all must attain to a higher standard than is reached on passing out of Dartmouth, and some, selected by a gradual process of elimination, will have to attain to a very considerable standard during a series of courses at Greenwich and elsewhere.

Now the course of Mathematics pursued in the public and grammar schools some half century ago was designed for the production of mathematicians. The average boy was put through as much of the course as he happened to be able to cover before dropping the subject, and the only justification to be found for such treatment in his case was that it furnished a mental gymnastic complementary to his studies in Latin composition, etc. Such Mathematics in its early stages is not of much use to the engineer or scientist, and in the effort to arrive early at the relatively advanced methods which are of use, the technical schools tended to sacrifice the teaching suited to those who would pass on to still more advanced work; a scheme of "practical mathematics" was thus evolved, of great value in economising the time of those whose requirements were strictly limited, and at least as intelligible and convincing to the ordinary student as the traditional course.

The problem to be solved is mainly one of administration. Three groups of cadets, each about 75 in number, enter Osborne every year; each group is divided into a number of classes according to ability in the several subjects studied, and these classes persist without considerable

change throughout the four years. So it was possible to design for the less able, who would not pursue Mathematics beyond the passing-out standard, a course of "practical mathematics" and to supplement it, in the case of cadets who could work more quickly, by the teaching required as a foundation for their subsequent work and desirable as a mental discipline. This supplementary teaching is directed towards the development of the power of abstraction and of deductive reasoning; in courses of practical mathematics every effort is made to keep in touch with the concrete, and to allow intuition and inductive processes, in which the generalisations are often rather bold, to take the place of deductive reasoning.

The supply of teachers competent to carry out such a combined course is at present not very large. A man must have received an essentially mathematical training if he is to realise the importance of sound foundations, and to lay them properly; success in this training demands a concentration of effort incompatible with the practical study of science or engineering which is essential for the teacher of practical mathematics. The plan adopted at Osborne and Dartmouth is to secure trained mathematicians and make all possible opportunities whereby they can develop a scientific habit of mind. It is not sufficient to ensure close contact outside the class-room between mathematician and engineer or scientist, nor to provide for courses, however extended, of "looking on" at science teaching, but a most marked effect is produced by arranging for mathematical masters to give science teaching themselves for a few terms both in lecture room and laboratory. The natural spirit of competition leads to the acquisition of the point of view which they see is effective in the case of science masters taking parallel classes or working with them in the laboratory. It is obviously desirable that these science masters should themselves be in touch with practical engineers to complete the chain. The confidence implied in the adaptability of the mathematical mind has been fully justified in our experience; I am satisfied that even if particular classes have suffered in their science work for a short time, the improvement in the mathematical instruction makes up for it to them, and of course the general efficiency is greatly increased.

There is another peculiarity in the education of naval officers, for which provision must be inade in their mathematical teaching. For five years after passing out of the training cruiser they have to serve at sea and concentrate their attention on learning the active duties of their profession. The only breaks in this period consist of short courses of teaching in purely professional subjects, and an examination chiefly on those subjects. The practice of this profession necessarily involves continual use of the theoretical knowledge acquired before going to sea; but organised teaching in Mathematics

is suspended, strong inducements being, however, held out to encourage these junior officers to study it for themselves. At the end of this period those who are fit to be specialists will attend courses involving Mathematics of an advanced nature, and it will be necessary for them to be able to refurbish in a reasonably short time what has grown rusty in the five years spent at sea. It is therefore desirable so to present Mathematics to them in the earlier courses that this fallow period shall be as little detrimental as possible. It is clearly inadvisable to aim at great skill in manipulation of symbols at that stage, for such skill rusts quickly and the whole process would have to be repeated. The aim must rather be to accustom them to rely on their own powers of working with the minimum of material in the way of standard types, processes and results. Although the special circumstances compelled the adoption of this method, there is little doubt that it is preferable in almost all cases, since for the majority of men life subsequent to school and college is a fallow period in this

sense.

This general principle of course admits of exceptions. For example, midshipmen get continual practice in the recognised forms for performing manipulations in navigation; and in other professional subjects every one makes use of standard methods for economising thought in operations so frequently repeated that it is less trouble to remember them than to think. But in the general teaching of Mathematics such economy of thought is out of place, except in producing facility in operations which are of frequent use in all walks of life.

The fear has often been expressed that a reduction of the traditional drill in manipulation, by which a knowledge of these standard types was instilled, would lead to a decrease in accuracy, already sadly to seek. It is not easy to bring forward proof or disproof of this view, but experience goes to show that other factors are more important in producing accurate work. First of these is age, or maturity of mind; it is largely a matter of the growth of a feeling of responsibility. Our forefathers relied on the birch to foster it; we have to find a substitute. A priori, this feeling of responsibility is not likely to be encouraged by a wearisome course of manipulation which carries no conviction of its value to a boy, while it is likely to grow when he is working at problems which are real to him, producing results whose accuracy he can himself check; and great advantage has been found from developing to the utmost the habit of checking results by independent processes. Again, the methods employed in the various branches of physical science admit of different degrees of accuracy of observation, and attention must therefore be paid to the degree of precision advisable or allowable in the computations based on the observations; this presents the question of accuracy in a light which impresses a boy since he is personally interested in the matter.