I am afraid that what I am going to place before you to-night will be somewhat incomplete. This is unavoidably so because I have not had sufficient time at my disposal to test my views thoroughly. The evidence which I have to offer, therefore, in support of my case is necessarily scanty, and it would be premature to force unwarranted conclusions from it. Moreover, what little testing I have done has been confined almost wholly to one special problem. On that account there is a danger that my presentation of the topics will appear to lack proportion. I hope, therefore, that you will bear these facts in mind when judging this paper, for otherwise your verdicts could not be altogether fair.

In the July number of the Gazette Mr. E. G. Phillips says: “Frequently it happens that students whose early training has been on the lines of many of the existing elementary text-books on the Differential Calculus come up to the University never even having heard of a differential! It would be of the greatest possible assistance to those responsible for the later teaching of the Calculus if the schoolmasters taught the subject from the differential standpoint right from the start”. I have the very greatest hesitation in subscribing to this latter statement.

Mr. Trevor Dennis, in opening the discussion, called attention to the proposed Syllabus f or Additional Mathematics for School Certificate as circulated amongst members, and said This syllabus has been drawn up by the Manchester Branch after consultation with other Branches, and we ask your help, your blessing and your advice as to amending it, so that we may put it before the Joint Matriculation Board of the Northern Universities as a syllabus which should stand as a separate subject. The Northern Universities School Certificate and Matriculation is a tremendous examination. Last July no less than 460 schools took the examination and 18,720 candidates were presented.

In talking about the relations of mathematics and culture or, rather, about the cultural value of mathematics, I think it advisable that I should first say a little about what I suppose the word “culture” to mean. In its general use it contains, I think, two ingredients, as it were. It refers to information and it refers to sensibility An ignorant man, however refined his natural sensibilities may be, is not regarded as cultured. And a man who has amassed an immense knowledge of facts, but is quite unable to distinguish between what is valuable and what is not valuable in his information, is not regarded as cultured. Of these two elements, I am inclined to think that sensibility is the more important. What the old writers on aesthetics used to call taste is, I think, more important than knowledge for the formation of a cultured outlook.

Whereas the works on Trigonometry, Geometry and Determinants were published over the name of Charles L. Dodgson, Student and Mathematical Lecturer of Christ Church, Oxford, the two small works on Logic—The Game of Logic and Symbolic Logic, Part I — and the two interesting logical notes in Mind appeared over the signature of the creator of Alice. Thus on the theory of double personality which the Rev C. L. Dodgson tried so hard to establish (and so unsuccessfully, since his was a completely integrated though singular personality), the formal works on logic are to be classed with those informal ones which are the delight of the civilised world.

Explanation of terms used. For “Bordered Squares” the term Heart is given to the core around which the border is to be arranged. The Heart can be one of several different kinds. In our case, the rows and columns do not sum to the same amount; but the Heart is Associated, that is, every pair of numbers equidistant from the centre of the Heart in a straight line sums to the same amount, here 50. The numbers which make up such a pair are Complementary Numbers. When the numbers of these pairs are subtracted one from the other, they give the Complementary Differences.

The reports of the London University on the General School Examinations throw an interesting light on the knowledge of candidates of the subject of approximations. The 1928-30 reports make it quite clear that the subject is badly taught and that glaring errors occur very frequently The 1929 report states that this fault is general and is not confined to particular schools. When it is considered that the University examines annually about 25,000 pupils from schools all over the country there is justification for the suggestion in the title of this paper.

In Sir Thomas Heath’s entrancing Manual of Greek Mathematics it is stated on p. 52 that the Babylonians discovered “a most perfect or musical proportion” between two numbers, which Pythagoras introduced into Greece. This is the ratio a ½(a + b):: 2ab/(a + b) b. The two middle terms are the arithmetic and harmonic means of the extreme terms. Not knowing of this when considering the fundamental principle of the logarithm, it seemed to me that the use of these two means should assist in evaluating the natural logarithm of a ratio in which the numerator and denominator are the first and last terms. In this way I found the surprisingly exact but approximate expression for the numerical value of such a logarithm which appeared in Nature of 14th March, 1931.

I do not know what proportion of the users of mathematical tables go beyond the use of proportional parts in interpolation work. But there are undoubtedly, on the one hand, some tables whose construction would be unnecessarily laborious and expensive were they to be at such an argument interval as to render linear interpolation feasible, and, on the other hand, there are those who find it necessary to use such tables. Given such a table, there are various methods of performing the interpolation. There is the use of the Taylor series, f (a + h) = f(a) + hf′(a) + ½h2f″ (a) + ..., which requires a knowledge of the derivatives of the function as well as the function itself. Then there is the use of the Lagrange formula of interpolation, fx = a0f0 + a1f1 + a2f2 + ..., involving a number of tabular values on both sides of the required value.

The discussion on symbolic operators in differential equations in recent numbers of the Gazette shows the need of a different method of attacking the linear equation of the second order with constant coefficients, which is the case most wanted in practice. It seems a pity that no method is given in the English books except the symbolic one, in which pass students get lost owing to their not understanding exactly what they can or cannot do with the operators. And is this to be wondered at when a theorem which gives erroneous results if not interpreted in a particular way has stood in Forsyth’s Treatise for forty years?

The study of Geometrical Optics has not attracted much interest or attention for some years now. at all events until recently, save only among a quite small company of specialists, and, as but one particular result of this, the divorce between optical theory and optical practice has been lamentable, as, for example, in that problem of so great practical importance, the design of the symmetrical optical system. For the one part, the subject has been regarded as isolated and of little theoretical interest, and scarcely susceptible of wide and profound generalisations, then, too, and this more especially from the point of view of university teaching, geometrical optics has seemed to stand apart from the path to a general mathematical education and training, and, in consequence, in several universities the study of it has been almost abandoned, save only in its most elementary aspects.

At the beginning of the seventeenth century π had already been calculated to 35 places of decimals by the Archimedean method of finding in succession the perimeters of inscribed and circumscribed polygons, each having twice as many sides as its predecessor.

But at this time several mathematicians, notably Snell, were seeking shorter methods of approximation, and finally Huygens in thirteen propositions proved or disproved the assumptions they made, and in three more propositions, making use of the known position of the centre of gravity of a parabolic segment and a theorem of his own on the centre of gravity of a segment of a circle, showed how to find a closer upper limit for π from any polygon than had yet been obtained, and by a seventeenth theorem “on centre of gravity”, of which he does not give the proof or even the enunciation, he found also a closer lower limit.

On the dashboard of a car we have a collection of direct-reading instruments, such as voltmeters, clocks, etc. The great simplification of practical physics which has been brought about by the ever-growing use of direct-reading instruments has unfortunately not been accompanied by an equal simplification of mathematical teaching. Schoolmasters ignore the dashboard, and base their teaching on Greek mathematics, for, though they profess to build on those foundations which are familiar to their pupils, they show a strange partiality for the familiar knowledge of two thousand years ago.

Among many English writers of text-books on arithmetic and algebra there appears to exist a state of confusion regarding the use of the sign of multiplication. Here are three instances. Godfrey and Price’s Arithmetic (1915) contains the following statement:

“The sign × means multiplied by; thus 3 ft. × 7 = 21 ft., and (7 × 3) ft. = 21 ft.; but 7 × 3 ft. is unmeaning ” (p. 13). Yet, on page 147, we find the expressions

in which multipliers precede the corresponding multiplicands.

The next International Congress of Mathematicians is to be held at Zurich from 4th September to 12th September of this year. The complete programme of lectures and sectional meetings has not yet been received, but the list of those who have already promised to speak is so strong that an attractive week is certain.

Besides the scientific arrangements, there are plans for social functions and excursions ; for these Switzerland should prove an ideal setting.

If the elements of a determinant of order n are given by

ars(r, s = 1, 2, ... n),

then the rule for determining the sign to be affixed to any term of the determinant is usually given in a form equivalent to the following arrange the n elements of the term with one set of suffixes in ascending order (i.e. 1, 2, ... n), and count the number of inversions, equal to i, say, in the other set. Then the correct sign is ( − )i.

1. One of the two curious self-reciprocal formulae stated by Rama-nujan,

where

of which I have recently given a proof, can be established by using a result proved by Stieltjes that, if x > 0, m ≥ 0, 0≤a≤ 1,

The proof only requires the above result in the case in which m = 1, a = 0. While seeking an extension of the result (1. 1), I have obtained an expression for the continued fraction (1. 3) in the form of an infinite series.

One of the chief appeals of Trigonometry to the beginner lies in the practical nature of the subject. The first aim in this respect is to obtain the formulae for the solution of the general triangle, but hitherto there has been much delay in the logical development of the subject up to that stage because of the need

(a) to prove the half-angle formula cos A =cos2 ½A − sin2 ½A and

(b) to show that cos (180° − A) = −cos A and sin(180°−A)= sin A.

The teaching of mathematics has been brought to so high a pitch of excellence by the eminent teachers of the past century that it may be thought there is little scope for improvement in their methods.

Since the radical revision of our geometry text-books towards the end of the nineteenth century, when modern methods first superseded the age-old system of Euclid, we find no fundamental change of doctrine until we come to consider the place of trigonometry and of the calculus in present-day curricula.

1. The foci, directrices and the eccentricity of the conic given by the general equation of the second degree can be determined by the use of the fundamental property of the conic, that SP = e . PM. Let the equation be

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0.