On some previous occasions when we have had schoolmaster presidents they have told us about the events of the early years of our Association.

Canon Wilson, who addressed us about twenty-five years ago, was an original member; he taught mathematics not far from here, at Rugby. Mr. Siddons told us of the beginning of mathematics in some well-known schools in the nineteenth century; he was himself a pupil of another of our original members, Rawdon Levett, who taught for thirty-four years at King Edward’s School in this city. Levett was our first secretary, and when he retired the president attributed to him a full share of the credit for our initial successes. He was a schoolmaster of genius: one of the greatest teachers of his time. It is natural that we should think of him this year when we are meeting in Birmingham. Two years ago, Mr. Bushell, the son of another original member, told us about the last century of school mathematics.

In Mathematical Note 1911, this journal, Vol. XXX, No. 290, A. H. MacColl forms the determinant Δ of a pan-magic square of order 4 and states that Δ = 0. He also remarks that “the property appears to be peculiar to pan-magic squares of order 4”. Below we give our demonstration of the above theorem and extend it to the determinants of all pan-magic squares of even order. We shall show that the determinants of all pan-magic squares of even order have zero as their value. In the proof that follows below we make use of a property of pan-magic squares of even order.

It is Well known that square roots of integers and rational numbers can be calculated by means of recurrence relations. Further, the same recurrence relations can sometimes be used to express a square root as a continued fraction. From a certain type of recurrence relation which gives a square root it is also possible to express that root as a rapidly convergent infinite series. As might be expected, the method is connected with the theory of continued fractions, and that connection will be given.

This suggests to find a point P which is the centroid of its antipedal triangle as to ABC.

Let LMN be the antipedal triangle of P and A′, B′, C′ the inverse of A′, B′, C′ with respect to a circle Γ having its centre at P. If J is the intersection of AP and B′C′, the inverse J′ of J as to Γ is the point where the parallel to MN drawn through L meets again the circle BPCL. Hence the ratio PJ: A′J equals PA: J′A, and this means that the barycentric coordinates of P are the same in the triangle LMN as in the triangle A′B′C′ Therefore, if P is the centroid of its antipedal triangle, it is also the centroid of A′B′C′

In the first part of this article we studied the relation between mathematics and the child and tried to suggest that a greater place could be given to mathematics in the schools by taking into account how the child learns. We tried to show that, to give the child a knowledge of mathematics, various approaches can be used, each of which can serve as a basis for the following one if we follow the development of logic, using the set of tools mastered at each stage in order to progress from one level to the next.

LET x be a real continuously variable quantity, and the functions Ft(t: 1, 2, .. ) be arbitrary functions of x which are assumed to have derivatives of sufficiently high order in a given interval of x. Let P≡d/dx so that P is skew-hermitian and satisfies

APRÈS avoir signalé un nouveau cercle associé à un quadrangle plan quelconque nous nous proposons d’étendre au quadrangle inscrit à un cercle des pro priétés du triangle et du tétraèdre signalées par G. de Longchamps, J. Griffith et par nous-même.

IT is well known that the identity

enables one to generate right-angled triangles with integral sides merely by putting x and y as integers (x>y). It is sufficient that x, y be integers, to give a right-angled triangle with hypotenuse x2+y2 and sides x2—y2, 2xy. Is the condition necessary? No, for the triangle with sides 9, 12, 15 (a multiple of the 3, 4, 5 case) cannot be expressed by the above identity without using surds. VVe will, however, prove the following proposition: