An arithmatical amusement, said to have been first propounded in 1881, is the expression in the expression in the ordinary arithmatic and algebric notation of the consecutive integers from 1 upwards, as far as practicable, by the use of four “ 4 ”. I have mentioned the problem in my Mathematical Recreations, but my friend Mr. Oscar Eckenstein has now carried the solutions considerly further: I think a bare statement of our procedure may be of interest, and perhaps some readers ot the Gazette may amuse themselves by filling in the details ot the analysis or applying similar methods to higher numbers.

As a first result of the labours of the International Commission on the Teaching of Mathematics, appointed in 1908 on the initiative of Professor D. E. Smith, the distinguished author of Rara Arithmetica, some 9000 printed pages were presented to the International Congress of Mathematicians at Cambridge. It is obviously impossible to go into details: I can only hope to mention one or two facts, and to give a general impression of the problems raised.

In the second edition of my Contents of the Fifth and Sixth Books of Euclid I have taken the ideas stated in the preceding article as my starting point, and built up the conditions involved in the 5th and 7th definitions in the following way:

1. 1. The ratio of a magnitude A to another magnitude B of the same kind is a real number, rational or irrational, determined in the manner explained in what follows. It is denoted by the symbol A : B.

2. 2. If the magnitudes A and B have a common measure G, so that A = aG, B = bG, where a, b are positive whole numbers, then A : B is defined to be the rational number .

1. It seems most convenient, in virtue of the well-established pedagogic maxim that “algebraic subjects should, as far as possible, be introduced as generalisations of known principles in Arithmetic,” to base our teaching of limits and convergence upon the arithmetic theory of “Approximations,” with which all scholarship candidates are familiar as a practical instrument.

The formal definition of a limit by means of a sequence is usually found by the beginner very hard to understand, but if presented from the point of view of approximations it is less difficult.

So much for preliminaries. We come now to the formulation of a postulate which may be compared to Euclid's “Let it be granted that a straight line can be drawn from any point to any other point ”; a postulate, by the way, whose significance has been much underrated by later geometricians. To explain fully the closeness of the analogy would involve the discussion of the cataloguing of groups higher than the first, for which I have not space here, but I may indicate its general nature, to those familiar with the subject, thus: In “projective” geometry the existence of straight lines is assumed, and from them, by means of the Quadrilateral Construction, we obtain an unique method of determining any number of points in a straight line, with reference to three given points in it. My procedure is the exact converse of this. I assume what corresponds to an unique method of determining any number of points in a straight line, with reference to any three of them, and from that I deduce a method of drawing other straight lines, that is, if I am given a plane to draw them in.

Mr. H. D. Ellis has been good enough to compile the following statistics : A list of members of the Association on January 1st, 1911, classified according to locality, shows a total of 680, of whom England contains 531, Wales 21, Scotland 23, Ireland 14, Isle of Man 1, Channel Islands 2, South Africa 23, Canada 6, Australia 9, New Zealand 4, West Indies 1, Malta 1, India 17, Brazil 1, Egypt 3, Italy 1, Japan 2, Boumania 1, Syria 1, U.S.A. 18. The 531 members in England are distributed thus : London 146, Bedfordshire 7, Berkshire 13, Bucks. 12, Cambridge 31, Cheshire 6, Cornwall 2, Cumberland 5, Derbyshire 4, Devonshire 11, Dorset 1, Durham 4, Essex 11, GIos. 26, Hants. 19, Herts. 12, Hereford 1, Isle of Wight 2, Kent 23, Lancashire 25, Leicester 6, Lines. 1, Middlesex 8, Monmouth 1, Norfolk 8, Northants. 3, Northumberland 4, Notts. 3, Oxford 13, Rutland 4, Shropshire 6, Somerset 8, Staffs. 4, Surrey 21, Sussex 18, Warwickshire 13, Wilts. 2, Worcs. 9, Yorkshire 38.

In its most simple and general form the proof by recurrence of a formula containing constants and one variable consists in giving a number of successive integral values to the variable, and showing by trial that the formula is true for those values. The next step is to show that the form of the result remains the same when n+1 is substituted for n, or that each result depends upon the preceding one. If this second step is neglected, as it often used to be, the proof seems to be invalid.

This proof by mathematical or successive induction has been very generally accepted by mathematicians and neglected by logicians.

(In reference to A. C. Dixon’s note on lm + 2m + 3m +...+nm, on pp. 283-4 of this volume, the following further notes may be of interest. They are taken, with some alterations, from a paper communicated to the London Mathematical Society in February, 1910.)

1. Giulio Carlo dei Toschi di Fagnano, Count and Marquis, was an Italian mathematician of the first half of the eighteenth century. He was one of the second generation of the followers of Leibniz, a generation rather pale and unimportant compared either with the first great founders of the Differential Calculus or with the third generation that dated from about the middle of the eighteenth century, of which Lagrange was the leading genius.

Several of the men of this second generation are known mainly for some one piece of work. This is particularly the case with Fagnano, who extended the application of the Integral Calculus to the rectification of curves in a way that may be regarded as the first germ from which afterwards sprang the Theory of Elliptic Functions.

Experiencing the well known difficulty of rectifying, by elementary integration, many curves whose equations seem, at first sight, easy to deal with, he showed that on some curves two arcs could be found whose difference was rectifiable ; although each separate arc presented the same difficulty as before.

Arithmetic has been until recently the only branch of Mathematics generally taught in the Primary School; and as this subject deals exclusively with numbers in their simplest and most practical form, it must, in the order of teaching, be the first introduced to the child. The premier place must also be awarded to it when considering the relative importance of the various branches composing the unity of Mathematics.

1. Let AP, BQ, CR be three lines through the vertices of the ΔABC which rotate with equal angular velocity ω, and which make equal angles with the opposite sides, the initial position being that of the three altitudes of ABC. Then the three circles AQR, BRP, CPQ, which meet in the well-known point O, will be shewn to form three coaxal systems; and among other things we shall shew that the locus of O is the circle on GH as diameter, G being the centroid and H the orthocentre.

1. The present paper is an attempt to sketch a mode of procedure for teaching Limits and Convergence to beginners. It is felt by many teachers that at present

(1) the subject is introduced too academically;

(2) that its Arithmetical foundation is not brought into sufficient prominence;

(3) that therefore pupils leave school with only a very hazy notion of what it is all about;