In the preface to his Principles of Geometry, Professor Baker states that the “Foundations” volume “assumes only those relations of position, for points, lines and planes, which, furnished with a pencil, some rods and some string, a student may learn by drawing diagrams and making models.” When the beginner attempts to make these diagrams, it is not unusual for one or more points to go off the page. To illustrate how, in an important set of cases, it is possible to get all of the diagram on the page at the first trial is the purpose of this note.

The method of introducing e so common in text-books based upon an algebraic proof of the exponential series may be satisfactory for mathematical students, but it does not meet the needs of technical students (especially evening students), for it is impossible, if even desirable, to treat the matter along these lines in a manner at once brief, exact, and comprehensible. Moreover, it suffers from the great disadvantage that the property of e, which is of prime importance in technical applications, namely that ex is the function appropriate to the expression of the law of growth or decay so common in these—in other words, that ex is the solution of the differential equation dy/dx=y —appears as an afterthought, and not as the fundamentally important property, the raison d’être of the inclusion of e in the course and in the engineering formulae. The method outlined below, which has been tried for some few sessions, aims at presenting the differential equation as arising in the attempt to solve a concrete practical problem, and the discovery and identification of two solutions—one a series, the other in terms of indices. By obtaining the series first some idea of the use of non-terminating series, and appreciation of convergence, are gained. To simplify the numerical calculations, the index solution is then found. Here e appears as a limit, and to obtain its numerical value recourse must be had to the series.

In teaching beginners the elements of the calculus, the teacher’s first real difficulty arises when he has to present the differentiation of the logarithmic and exponential functions; these being a natural outcome of an attempt to complete the rule for xn for all values of n, including the case when n=−1. He either has to make the approach as in texts on “practical mathematics,” by differentiating the series for ex, with all the tacit assumptions that have to be made; or he is dependent on non-rigorous treatment of limits, necessary to show that (ax−1)/x tends to the limiting value logea; or to find solutions of dy/dx=y by some such process as is given in Lamb’s treatise on the Calculus, without the necessary proofs of convergence and differentiability of the series assumed; or he is bound to discuss convergence and continuity fairly fully, and give such proofs as Lamb gives, or the equally severe method of Hardy’s text.

1. A Solar eclipse occurs whenever the sun, moon and earth have their centres so nearly in line that the moon hides some part, or all, of the sun from an observer on the earth’s surface. In the former case the eclipse is partial, in the latter total. The relative motions of the three bodies are extremely complicated; but as knowledge of the apparent positions of the sun and moon is of importance for safe navigation, these are regularly computed in advance and published by the Admiralty in the Nautical Almanac. Since the Nautical Almanac tables are the data for eclipse calculations, it is essential to understand the method there employed for recording the positions of the heavenly bodies, and the meanings of a few technical terms.

Mr. G. Goodwill, in opening this discussion, said that in dealing with the subject of units in mechanics, it seemed to him to be a help, in the first place, to recognize that, in general, when expressing the value of any quantity, the unit employed belonged to one of two kinds, determined by the business in hand. It would be either (a) a unit represented simply by a well-recognised amount of the quantity itself, or (b) a unit which, in the simplest possible way, took account of the relations of the quantity with others with which they might at the same time be concerned.

In the traditional text-book presentation and evaluation of logarithmic, exponential, circular and hyperbolic functions there is a lack of uniformity of method, and the demands on the analytical powers of the average student are considerable, if not excessive It has been shown in The Mathematical Gazette of June, 1926, that logarithmic and exponential functions may readily be evaluated by ab initio arithmetical methods. Incidentally it may be noted that the theory of the method is independent of the theory of indices. The definition of loge a is virtually and leads to a simple assignment of meaning to irrational and imaginary powers of a number.

We take two bodies P Q which have straight world lines OP KQ. This means in classical language that P and Q have a constant relative velocity. Our object is to find the configuration which an observer U reckons that of nearest approach of P and Q, and to find U’s measure of the least distance.

In his very interesting article on Asymptotes in the May Gazette, Prof. Nunn points out that the mathematician often asserts that a line always meets a conic in two points which may be at infinity, and that teachers and text-books dwell too little upon the fact that the word “point” is used in widely different senses.

Before proceeding to discuss certain aspects of modern atomic theory, I may perhaps be allowed to say a word as to the object of atomic theories in general. Most of us will, I suppose, agree that the object of physical theories is to set up a conceptual scheme which shall make it easier for us to classify and describe the experimental results in which we are interested, by providing, among other things, criteria which shall enable us to distinguish the essential from the unessential, and a scale of values which shall give us help in selecting for investigation certain problems from the infinite field before us. Every important theory is bound to bring into prominence research on certain groups of phenomena at the expense of research on other groups of phenomena: thus the general result of the electron theory is that in any physical laboratory three men out of four are working with vacua, and we see them staring intently at glass tubes with nothing, or very little, shut up in them. An immense number of beautiful results have come from these researches, exhibiting that order which is the true object and delight of the natural philosopher. A few experiments occasioned the original theory: the theory has provided the incentive for a vast number of further experiments, by suggesting for them a significance which would otherwise be lacking. The problem of the connection between radiation, matter and electricity occupies the thoughts of many of the best brains in physical science. This is a result of the electron theory; and this concentration on one problem inevitably draws attention away from other problems. Relatively little is being done, for instance, on the theory of solids, especially in the great problem of cohesion, not because it lacks interest or importance—it has intrigued natural philosophers since scientific speculation began—but because there is no simple, mentally manageable theory to guide us.

The following, written at the reguest of the Editor of the Gazette, gives an account of the mathematical part of the remarks made by the lecturer at the Annual Meeting of the Association last January. There are added some words dealing with geometry on a sphere, in regard to which some questions were raised at the meeting.

In the Memorandum from the Girls’ Schools’ Committee of the Mathematical Association, published on pages 13-15 of the Mathematical Gazette for January, 1926, we are told that the Committee deprecates the use of the term “inferiority” as applied to girls’ mathematical work by the Report on the Differentiation of Curricula between the Sexes. Further, the Girls’ School Committee says: “Where boys and girls are given equal opportunities, as in some mixed schools, the consensus of opinion is that very little difference in capacity between boys and girls is shown in work up to Matriculation standard….” It seems desirable that the question should be put on a sound basis, and with this in mind the writer has endeavoured to draw on material available to him for a statistical inquiry into the points involved. For the last two years he has been engaged in teaching in a mixed school where the boys and girls are taught mathematics together by the same teacher in the lowest two forms, and the forms are chosen on a basis of general standard in school work. The marks are given below. In each case the marks are those obtained by equating the quartiles (by a modification of a method due to Prof. Nunn and given by the writer in the Forum of Education in papers in 1924 and 1925) to 40 and 60, in homework, weekly tests, and terminal exams. The two marks given are those obtained at the ends of the Christmas and Midsummer terms.

In all general equations in mechanics the fundamental units, length, time, mass, enter homogeneously; the result is that these equations are true whatever the units of length, time, and mass may be. Angles, however, occupy an anomalous position. An equation which involves angles, angular velocities or accelerations, is true only for a certain particular unit of angle, usually the radian. Thus the formula T = 2π/ω, for periodic time with angular velocity ω, is only true when the angular unit is the radian; so also for arc = rθ, or for . The reason is, that these equations are not homogeneous in angle; π, sin θ, and cos θ are pure numbers, arc and r are lengths. In explanation it is commonly said that angle is of zero dimensions in length, time, and mass. So it is, and yet it is not a pure number, but a quantity just as much as length or any other concrete magnitude. If π were the symbol for a certain angle, instead of being a pure number, equations like T — 27π/ω, α+β + γ=π, etc., would become homogeneous, and true for all angular units, π would then resemble the symbol g, which stands for a particular acceleration, 32.2 ft./sec.2 or 981 cm./sec.2, etc.; but π has too long been identified with the number 3.14159… , and is too useful in all sorts of questions not involving angles at all, to be appropriated in this way. If, however, we introduce another symbol, say P, for the angle π radians or the angle of half a revolution, we may make all our equations homogeneous in angles, and restore their generality; it will do equally well, and be generally simpler, if instead of P we use the symbol K, denoting an angle of one radian.

Descartes at the beginning of his work on Geometry (1637), which would now be called a treatise on Algebraical Geometry, in order to shew the relation between Algebra and Geometry considers four types of quadratic equations. It will be noticed that these are special forms, the constant term of a quadratic equation being as a rule, not a square, but the product of two lengths, and necessitating the use of Euc. VI, 13 to find the equivalent square.