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.

Pierre-Simon De Laplace was born on 23rd March, 1749, at Beaumont-en-Auge, a village four miles west of Pont l’Évêque in Normandy. As its name would suggest, it stands on high ground : and from an old Benedictine Priory where Laplace went to school, only a stone's throw from this birthplace, there is in fine weather a superb panorama, reaching to the English Channel seven miles away.

The original documents relating to Laplace's life have almost all perished in quite recent times: most of them as a result of a fire which in 1925 destroyed the château of Saint-Julien de Mailloc, near Lisieux, the home of his great-great-grandson the Comte de Colbert-Laplace : some had been destroyed earlier, when his house at Arcueil near Paris was looted by housebreakers in 1871 : and some were lost very recently in the bombardment of Caen. This is the more regrettable, because many of the statements regarding Laplace in the histories of mathematics are false. His father, Pierre de Laplace, owned and farmed the small estate of Mérisier near Beaumont, and held the office of Syndic of the parish. When Laplace became a Marquis, it was the fashion to regard him as having risen from a very humble origin: but the fact was that he came of a country family of good social standing One of his father's brothers was a priest and one a surgeon, and his greatuncle, Maître Olivier de Laplace, had held the title of Chirurgien Royal.

Most of the standard textbooks and many teachers of Applied Mathematics make use of a velocity-time graph to facilitate the solution of elementary kinematical problems including vertical motion under gravity, but it is unusual to extend the method to projectiles and allied problems, in spite of the ease with which the more usual type of problem can be solved. Such problems may be solved by simple calculations based upon the properties of similar triangles and the areas of rectangles, triangles and trapezia, that is, by the normal velocity-time graph procedure.

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.

The most surprising thing about this formula is its use of the Bernoulli numbers, and it is natural to ask why they appear. The answer is that it is not the B’s which insist on entry, but the numbers Ar = Br/r!. We use the notation where all other B with odd subscripts vanish, and those with even subscripts are alternately positive and negative. We shall often ignore the fact that A3, A5, .. vanish, thereby refusing to complicate formulae unnecessarily.

The following theorems are closely connected in various ways; one common thread that runs through them is that in each a critical stage of the proof consists in filling in a missing value in the range of values for which the result is known to hold. Another common feature, of no logical importance but none the less the raison d'être for this article, is that each serves to illustrate important fallacies and omissions in current accounts of elementary algebra and analysis.