(2) He knows that to add to or subtract from the index the integer n multiplies or divides a power of 10 by 10". = From (1) he can draw a graph of y 10, for values of x from 0 and 1, and deduce the logarithm curve x = logio y. From this curve and (2) he can find, roughly, the logarithms or anti-logarithms of any number. E.g., if we require log 40 or log 0 04, we find from the curve that log 4 06, and from (2) deduce that log 40 16 and log 0·04 = 2·6. = = We can also deduce from (2) the rule that if the first significant figure is n digits to the left of the unit digit the characteristic is n, if n digits to the right of the unit digit, it is n. The rest of the chapter can be taught in the usual manner. (16) Computation.-Owing to the fact that we have not a decimal system for our coinage or weights and measures, questions are bound to occur which involve complicated methods of computation. Some of these can be solved most easily by the method of Practice, see page 230, but no general method can be given. In fact, the most difficult thing to learn is the choice of the most suitable method in a given case. A few typical questions are given below: (a) Find the value of 2,560,000 milreis, given that 1 milreis 48. 104d. (b) What is 3,7651. worth in American money when the exchange is 4 dollars 87 05 cents to the pound? (c) Find the simple interest on 12,6251. for 17 days at 34 per cent., i.e., find the value of 12,6251. × '17 34. X (d) A rate of 7s. 14d. in the pound on a parish produces 3,216l. 13s. 7d. What is the rateable value of the parish? (e) The moon makes a complete revolution among the stars in 27 days 7 hrs. 43 mins. 11 54 sec. Find the angle through which it moves in a day, to a hundredth of a second. These are doubtless extreme cases, but they are all examples that may occur in actual affairs. It is not desirable that a boy should spend much time on the solution of such questions, but after he has finished doing regular work in Arithmetic an occasional question of this kind may be found good practice. It is suggested that a chapter should be devoted to miscellaneous methods of computation, that the first section should be given to an explanation of the method of practice, and that this should be followed by a collection of miscellaneous examples involving a variety of methods. The best position for this is perhaps after the chapters on logarithms and approximate work. (17) Simple Interest, Compound Interest, Discount, Stocks and Shares, Exchange Operations.-For the general question of Commercial Arithmetic see p. 239. The following suggestions are made on the assumption that it is taught to the same extent as at present. As far as "Simple Interest" is concerned, it would perhaps be best, from a mathematical point of view, if after the technical terms had been explained, only one set of miscellaneous examples were given. Such questions are either direct questions on percentages or questions to be solved as algebraical problems or by a formula. Similar remarks apply to discount. Some questions on simple and compound interest should be set as examples of arithmetical and geometrical progressions. The factor method of dealing with compound interest should not be neglected. E.g., the amount of 5001. in n years at 4 per cent. C.I. is 500l. x (104), the C.I. is 500l. { (104) " — 1 } { &c. In the section on stocks and shares, the writer finds it convenient to deal only with shares when teaching the meaning of the technical terms. After that, stock can be introduced and with it the further difficulties connected with the ideas of capital and revenue. These are very real difficulties. The arithmetical side of exchange operations has been neglected in recent years. It seems hardly worth while to load the Arithmetic text-book with an account of the practical details of this subject, which, without being treated very fully, may easily occupy sixty pages (see Jackson's Commercial Arithmetic-Macmillan), but it is a fruitful source of examples in computation. As all this work is only partly mathematical and is in addition an explanation of the details of modern business, it is necessary that the latter should be as accurate as is possible with a given limit of space. To take an example from discount. The meaning of the true discount and bankers' discount of the text-books has been explained on p. 233. Of these two, the former is treated as the more important in nearly every text-book, the latter is dismissed as of slight importance. The writer has made inquiries, and can find no example of "true discount" in actual practice. Discount is the term applied to the sum deducted from a debt due at a future time in consideration of its payment before it is due. This sum is always calculated by the method of "bankers' discount." Of course, the idea of present value occurs in modern business, e.g., in actuaries' work, but it is connected with the idea of interest, not of discount. "True discount," reckoning compound interest, is still further removed from actual practice. It is also desirable in questions on interest and discount that the data should be reasonable. In a set of examples on discount I find that the average time after which a bill becomes due is over four years! This must give boys a curious idea of the credit system of the country. (18) Graphical Work.-It is assumed that graphical work has been used throughout as a means of illustration, and occasionally, as in the chapters on proportion and on logarithms, as a means of calculation. It is, however, customary to group together in one chapter examples of various kinds that lend themselves to graphical solutions. Such methods need not be artificial. If a series of values have to be changed from one unit to another, e.g., from feet to metres, and no great accuracy is required, a graphical solution would perhaps be the simplest. The complicated time-tables of some of the railway companies are worked on squared paper and allow the position of a train at any time to be seen at a glance. Many questions of interpolation cannot well be solved by any other elementary method. (19) Problems.-Little need be said of these, except that their value is much diminished if they are separated into types. This has been discussed on page 234. A great deal of space is devoted to problems in many books. All that seems needed is a set of miscellaneous examples, preceded by a few worked-out examples carefully chosen. There is a tendency in some modern books to overload the Arithmetic text-book with applications to matters which, though of undoubted interest and value, do not belong essentially to the Arithmetic course. I refer to such subjects as statistics, permutations, surveying, chemical arithmetic. The writer is of opinion that the sections given above include the essential parts, and that too great an amount of detail gives a boy unnecessarily the impression that Arithmetic is a more difficult subject than is actually the case. SUMMARY. Bringing together the results of the foregoing criticisms and suggestions, we may say that the chief aims which are generally agreed upon by those who wish to make considerable alterations in the teaching of Arithmetic are the following:---The reformers would clear away entirely many parts of the subject which do not seem essential, and also clear away elaborate and artificial developments of many other parts. This pruning of the subject would enable the student to begin Trigonometry or some other branch of Mathematics at an earlier stage. But part of the time set free by the reduction and simplification of the subjectmatter of Arithmetic could be utilised in teaching more thoroughly the important parts. For one thing, a much higher standard of work in straightforward computation is needed. throughout the country. The methods now employed are often clumsy and the general standard of accuracy is not high. It is perhaps utopian to expect to secure any uniformity of method even in the fundamental processes, but at any rate considerable improvement of method can be introduced and a higher standard of speed and accuracy secured. It will be a very great help towards this if pupils can be made to acquire the habit of constantly checking their work by rough approximation. In matters beyond mere computation the important principles and ideas (and they are not numerous) should be introduced at first more simply and thoroughly. Rules should not be used until a considerable effort has been made to explain their meaning and, if not to prove them, at any rate to make them seem reasonable by considering particular cases. Sets of examples should be so varied that they cannot be worked by a purely imitative process but must necessarily require thought. New ideas should be made as real as possible by concrete illustration. This should include geometrical illustrations-diagrams, graphical work on squared paper, &c., and also laboratory work in close connection with the mathematical teaching. Although it is not desirable as a rule that any time should be spent on the theory of errors, it is important that the pupil should have some appreciation of the fact that the degree of accuracy possible in an answer depends on the accuracy of the data. He should be able to give an answer roughly to that degree of accuracy that is justified by the data. Questions should also, as far as possible, be taken from the arithmetic of everyday life, and at any rate they should not offend one's common sense. In all these changes examining bodies can do a great deal to help or hinder. There are many reforms that could be introduced to-morrow if it were not for the demands of examining bodies. The examination papers in Arithmetic of the Civil Service Commission have improved greatly during recent years, and at present not only allow but encourage the teaching of Arithmetic on satisfactory lines. But the majority of the examination papers set at the present time are an obstacle to many changes which nearly all teachers regard as beneficial and are anxious to introduce. So long as these papers remain unaltered, the hands of teachers are tied and progress is necessarily slow. G. W. PALMER. THE EDUCATIONAL VALUE OF GEOMETRY. 66 Every great study is not only an end in itself, but also a means of creating and sustaining a lofty habit of mind; and this purpose should be kept always in view throughout the teaching and learning of Mathematics."-BERTRAND RUSSELL. The title of this paper has been chosen to indicate that the discussion will not be concerned with the value of Geometry as applied to other sciences or to practical ends, nor even with its place and importance in schemes of mathematical education. The purpose is to state the reasons which appear to have led to the universal acceptance of the subject as a necessary element in education, to ascertain to what extent geometrical teaching in this country can find justification in them, and to give some slight account of experiments in teaching made on this basis by the writer and his colleagues at Tonbridge School. Lest it should be thought, however, that this avoidance of the practical importance of the subject and its relation to other branches of knowledge imposes unreasonable limitations, it may be well to state the reasons for it. The danger of giving undue importance to considerations of practical utility need hardly be enlarged upon, since it is not proposed to consider Geometry from this point of view. I am more concerned to point out that if the advocates of the subject rest any part of their case on such considerations they at once enter into competition with a host of other interests, many of which have, on such grounds, much higher claims. The parents of a boy who is to adopt a business career will rightly prefer, if his education is guided by his future requirements, that he should spend his time on geography or economics, arguing that surveying and bridge building can have no relevance to his future interests; while those who take a wider but, still utilitarian, view will insist that subjects such as the chemistry of food and civics have stronger claims to a place in the education of every child. Still more dangerous is the plea that every educated man should have some idea of a subject of such wide utility. Apart from the claims of many other branches of knowledge, this has a further demerit in that the object of teaching the subject is implied to be the acquisition of encyclopædic knowledge, rather than the development of the mental faculties. The old conception of education as the acquisition of information is dead, and it least becomes mathematicians to do anything to revive it. The use of justifications of this type, even though it be only in secondary positions, is likely to defeat the aims of those. who advance them and to do much harm to educational ideals. A 13913 R |