Mr. J. T. Combridge, in opening the discussion, said. My first duty is to give you a brief summary of the Report on Entrance Tests and Initial Degrees, adopted in June 1930 by the Council of the Association of University Teachers. My second duty is to offer such comments on it as may provoke a heated discussion among those who might otherwise be disposed to agree in approving or condemning it. For the benefit of those who have a copy of the Report, I would say that the mysterious reference at the foot of p. 4, “See pp. 4-10”, in reference to the English test, is due to the fact that the Report is an offprint from a certain number of the Universities Review, and the test referred to appears on pp. 4-10 of that number

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 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.

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.

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.

1. In a recent paper Mammana considered the equation

z″ + pz′ + qz = 0,.....(1)

where x is the independent variable and p and q are functions of x, real, finite and continuous in the interval (a, b), and investigated the possibilities of its decomposition into the form

(D + α)(D + β)z = 0,.....(2)

where D ≡ d/dx and α and β are functions of x.