26. In equal circles (or, in the same circle), (i) if two chords are equal, they cut off equal arcs; (ii) conversely, if two arcs are equal, the chords of the arcs are equal.

27. Equal chords of a circle are equidistant from the centre; and the

converse.

28. The tangent at any point of a circle and the radius through the point are perpendicular to one another.

29. If two circles touch, the point of contact lies on the straight line through the centres.

30. The angle which an arc of a circle subtends at the centre is double that which it subtends at any point of the remaining part of the circumference.

31. Angles in the same segment of a circle are equal; and, if the line joining two points subtends equal angles at two other points on the same side of it, the four points lie on a circle.

32. The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

33. The opposite angles of any quadrilateral inscribed in a circle are supplementary; and the converse.

34. If a straight line touch a circle, and from the point of contact a chord be drawn, the angles which this chord makes with the tangent are equal to the angles in the alternate segments.

35. If two chords of a circle intersect either inside or outside the circle the rectangle contained by the parts of the one is equal to the rectangle contained by the parts of the other.

Proportion: Similar Triangles.

36. If a straight line is drawn parallel to one side of a triangle, the other two sides are divided proportionally; and the converse.

37. If two triangles are equiangular their corresponding sides are proportional; and the converse.

38. If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

39. The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle, and likewise the external bisector externally.

40. The ratio of the areas of similar triangles is equal to the ratio of the squares on corresponding side.

COURSE IN MATHEMATICS FOR MUNICIPAL SECONDARY SCHOOLS,

This paper offers for consideration a suggested course of Mathematics which the writer considers suitable for his own school, but as the type of school is by no means an isolated one and probably is becoming yearly more common, it may be found. of interest and possibly useful or suggestive to others. The school is a Municipal Secondary School of about 300 boys, the majority of whom are between 11 and 17 years of age; a few are above the higher limit, none below the lower. Very few boys proceed to the older Universities of Oxford and Cambridge, a fair proportion, probably four or five annually on an average, proceed to the local University; a good proportion remain till 16 to 17.

Of those who leave a considerable number enter engineering. firms as non-premium apprentices, or smaller manufacturing firms; a fair number enter as junior chemists into chemical manufacturing firms, e.g., coal-tar distilleries, gas-works, alkali and acid works, &c. ; some into surveyor's and architect's work, while a number inevitably find their way into the clerical work of banks, insurance offices, &c. Of those who leave it is found that a large number continue their education, chiefly in the technical school, but also in various commercial classes. A fair number while at school take the London Matriculation Examination chiefly in order to qualify themselves for future Degree work (probably 9 or 10 annually on an average), a few-two or three annually-take the Intermediate Science, either of the London or of the local University. This preface may enable the reader to judge of the class of student for whom the course is intended, and to what extent it resembles or differs from those in which he is more particularly interested.

In such a school, therefore, there is no need to consider the possible candidate for a mathematical scholarship at Cambridge and to subordinate the general curriculum to him. It is probable that in many schools this has been done with ill effects; the majority of the pupils who leave the school having mastered as a result an incomplete and comparatively useless and lifeless portion of a course intended to proceed to a higher level. They never reach the point at which they can utilise their knowledge, either for the purpose of obtaining pleasure for themselves from its use, or for the purpose of assisting their progress in other work in which they are interested. As a consequence, on leaving school their Mathematics falls into disuse, is neglected, and soon suffers from the atrophy to which sooner or later all knowledge. which has not reached the useful stage is liable. To be of any real good it is necessary that a subject studied should be carried to the useful stage-whether the subject is classics, modern

languages, history, science, &c., or mathematics-and it is a painful fact that to many pupils who leave secondary schools Mathematics, with the exception of quite elementary arithmetic which possibly they learned elsewhere, is not useful. By "useful" knowledge is, of course, not meant capable of being materialised into money, of assisting the wage-earning ability of the possessor, but " usable" knowledge, i.e., knowledge which the possessor can apply either for his own enjoyment, or that of other people, or for assisting in obtaining such knowledge. There are, of course, many grades of such usefulness from that of mere simple arithmetical knowledge of numbers and the ability to add and multiply, up to the ability to advance the domains of Science and Mathematics by independent mathematical research, but whatever be the extent to which the study of Mathematics is carried care must be taken that the work done shall result in useful knowledge, and that the student shall be able to utilise his Mathematics, to handle it as a tool, and be able to apply his knowledge, however elementary or advanced it may be, to whatever work or study he is interested or engaged in. He should not be obliged after leaving school to attend classes in Mathematics either under the title of "Practical Mathematics" or "Commercial Arithmetic" in order to understand the applicability of the mathematical knowledge he is already supposed to have attained, although of course he may, and it is hoped he would, desire to attend classes to extend it, and carry it to higher levels. He should be able to do ordinary calculations and computations accurately and with reasonable speed and should be familiar with the various labour-saving devices of general utility, e.g., logarithms, slide rules, &c. Accuracy is an essential; it is useless to know how to use 2.723 751 273 logarithms in the computation of X. X 1 760 296.5 result is obtained incorrectly when they are used. The means by which this accuracy is to be obtained is perhaps a difficult question-some indication will be given later-but it is largely a question of age and the sense of responsibility with regard to the work as well as of interest in it. He should be able to solve accurately any linear equation or simultaneous linear equations, with any reasonable number of unknowns, and to solve any quadratic equation. He should have a clear and ready knowledge of the properties of the chief geometrical figures, triangles, parallelograms, circles and, though less extended, of the conic sections. He should be able to complete the solution of any triangle when sufficient data are given and should understand the methods of obtaining the necessary data in surveying, i.e., the theodolite, sextant, and plane table. He should have a clear knowledge of the conditions necessary for equilibrium of forces, whether parallel or otherwise, and have clear conceptions of elementary dynamics, and of the meaning of energy and its transformations. A knowledge of simple harmonic motion and its

if the

relation to uniform circular motion is desirable, together with the form of periodic curves such as the sine curve, and the effect of superposition of two or more such curves in different phase. He should have also-and this is important and should be attained even if other parts are omitted—a working, if elementary, knowledge of the calculus, sufficient for the differentiation of simple functions, sufficient also to check integrals met with in physical, chemical, or engineering work by differentiation, capable of utilisation in questions of maxima and minima, and for the determination of areas, volumes, centres of gravity and moments of inertia. He should also be able to understand a graph, to read fully its significance, and to form an idea of the probable form of function indicated. The idea of functions and the interdependence of the two variable quantities in a graph is an idea of such fundamental importance and such far-reaching utility that it should be instilled from quite an early period of mathematical work.

It is of course almost inevitable that the objection will be raised, that in the above outlines of the desiderata of the mathematical training no consideration has been taken of the educational aspect of Mathematics, of its efficacy as mental training, as discipline and as "education" in its true sense, i.e., the widening of the mind. This objection is, however, in the writer's opinion, one of no weight. It is only necessary to draw attention to the lack of initiative, the feebleness of resource, and the stunted mental condition of the average pupil fed on the ordinary mathematical pabulum, to show that some change is necessary, and that the orthodox mathematical training has not inspired sufficient confidence in its educative value to be regarded as permanent. Further, the educative value of a subject does not depend so much on the work which is covered and the knowledge which is attained as on the methods adopted for attaining this knowledge, and it is primâ facie probable that the methods employed to render any acquired knowledge useful will be more truly educative than those which merely aim at the acquisition of knowledge with no thought of its utility. Further, the writer believes that work of the type of interpretation of a graph, determination of a maximum by calculus, or the consideration of the solution of an actual problem by, it may be, the united aid of trigonometry, geometry and calculus, is far more stimulating and educative than the laborious algebra of complex and continued fractions, long G.C.M.'s or artificial and disguised quadratics, which, in spite of the protests of mathematical teachers, still figure in many examinations, and are still a source of irritation to the clever, and despair to the slow boy in our schools. An objection is sometimes raised to the style of mathematical teaching here suggested, that the work is made too easy, that modern methods and modern schemes do not give difficulties to surmount, and that the effect is to make boys mentally flabby.

The objection does not appear to be a real one; there are sufficient real difficulties to be overcome without fictitious ones; these real difficulties need not be shirked and they can supply all necessary stimulus, but it is our duty as teachers to ease and not to create difficulties, to smooth the road and not to lay obstacles upon the track. If a thing can be done easily it is wrong to let students flounder through long and difficult methods; further, it is far harder, as also it is more important, to persuade boys that Mathematics is easy than to let them believe it is hard. Doubtless also some will object that the scheme of work suggested cannot be covered by boys at the age of 16 to 17. These objectors probably overlook the very large amount of time which is spent, it might be said wasted, on work of a highly artificial character, work which seldom or never is related to the actual demands of practical, physical, engineering, or commercial problems. I believe, however, that those who work on the lines suggested will find that the work can be done, and that it will prove more stimulating and interesting, alike to teachers and taught, than the wearisome and stodgy Mathematics through which many of us laboriously trudged.

Arithemetic and Algebra.-The two are grouped together because it appears unnecessary to divide them. Pupils should from the start be allowed to use whatever mathematical tool is required for the solution of the particular problem involved; the same also holds good for geometry, trigonometry, &c. Most of the pupils entering schools of the type considered have probably come from elementary schools. They are for the most part completely ignorant of algebra, but have a fair knowledge of arithmetic. They are capable of doing ordinary questions in multiplication and division of money, or of the ordinary weights. and measures, are familiar with the unitary method and simple fractions, their work is fairly accurate but is painfully slow and laborious, and very rarely are they in the habit of checking their answers, either by rough checks at the outset or by testing whether the result obtained is in accord with the statements made in the question. In decimals they are usually distinctly weak. Work can therefore start on the assumption that decimals are not known, and much work, both arithmetical and practical, is requisite in order to familiarise the pupils with their use. Since at the same time measurement work, chiefly in the metric system, is being extensively done as the initial work in physics, a great deal of practice in the use of decimals is obtained, and efforts are made to get pupils to estimate in decimals rather than in fractions, i.e., a length is estimated at 6:4 cins. not as about 6 cms., or as 18 37 cms., not as nearly 18 4 cms., and by this means pupils get to realise that 37 is less than 4, a fact which often causes some difficulty to the pupil whose Mathematics is completely divorced from practical work. Here incidentally it may be remarked that the practice of estimation appears to be neglected too much in the science work of schools,