A theorem of Van Aubel, which first appeared in concerns the figure obtained by erecting squares on the sides of a quadrilateral, and considering various line segments such as the joins of the centres of opposite squares. In parts of the theorem are extended to the case of (i) four similar rhombi, and (ii) four similar rectangles erected on the sides of a quadrilateral. Further, these two results are described as in some sense ‘dual’ to each other. Here we show that they are in fact both special cases of a further extension of the original theorem (see Proposition 6, and the remarks that follow it), and we also find several extra squares in the original Van Aubel figure (Proposition 11).

Some seven years ago an attempt was made to tell something of the story of Francis Walkingame, the author of The Tutor’s Assistant, which was first issued in 1751 and became the most popular arithmetic text book for over a century afterwards. Recent contributions to the Gazette shows that this author and his famous book are still unknown to many readers. Interest was aroused by Dr. T. M. Flett’s description of Edward Winder’s Exercise Book, which was clearly based on an edition of The Tutor’s Assistant of about 1830. In his description of John Stewart’s Exercise Book, written at Dundee some thirty years earlier, Mr. W More remarked that the rules were the same as Winder used and quoted additionally the Rule of False in very similar words to those used by Walkingame. This example is interesting as showing that Walkingame’s book was used in Scotland at the turn of the century. Richard Hartley’s 1846 letter gives a list of the later sections in Part I (Integers) of Walkingame and shows that he had just started Part II on Vulgar Fractions.

In a great many problems in probability, one is given one set of quantities Xj expressed in terms of a second set of quantities Yk, and one needs an expression for the Yk in terms of the Xj. Basically, of course, this is just a problem in solving equations, but the results can be expressed in a particularly elegant form when the coefficients involve the binomial quantities . This situation is one that occurs very frequently, and the resulting formulae simplify the derivations of many identities. The present expository account is intended to collect together various results in a convenient form. We shall need to recall briefly a few of the main properties of the binomial coefficients.

The general problem. This paper deals with particular cases of the following problem: A rectangular array of square cells is given; a particle P moves from one cell to another in such a way that, when it is in any cell, it is equally likely to pass across any of its edges. P is liberated in any given cell, and it is required to find the probability that it will ultimately emerge across any stated edge in the boundary of the array.

On the reverse side of the Moon, just south of the Sea of Moscow, is a crater named Tsu Ch’ung-Chih. Who is Tsu Ch’ung-Chih? What did he do to be thus admitted to the honoured company of Maxwell and Hertz, of Mendele’ev and the Curies, of Lomonosov and Tsiolkovsky?

Tsu Ch'ung-Chih was a Chinese mathematician-astronomer who nourished in the fifth century A.D. He made notable contributions to the calendar calculation and determined several constants with remarkable accuracy. For example, he gave a value of 27·21223 days for the length of the nodical month, the modern value being 27·21222 days. As another example, he found that the planet Jupiter completes seven and one-twelfth circuits of the heavens in every seven cycles of 12 years; this corresponds to a sidereal period of Jupiter of 11-859 years, which differs from the modern value by only 1 part in 4000.

The simplest overhand knot, if tied in a strip of paper and carefully pressed flat, forms a regular pentagon (Fig. 1). All higher order polygons can be produced in this way with the single exception of the regular hexagon which requires two strips. The reason for this exception is interesting.

The considerations which I propose to put before you to-day arose from an attempt, when teaching students, to build up the classical laws of Newtonian Mechanics inductively from laboratory experiments.

In this task we are immediately confronted with two fundamental notions, those of mass and force.

It will probably be admitted that no definition of a physical quantity can be accepted as adequate unless it implies a criterion leading to an unambiguous measure of the thing defined.

How big is n!? For example, to pluck a number out of thin air, what is the order of magnitude of 272! ?

This is the third note of a series on the factorial. The two previous notes [1] and [2] dealt with the factorial function x! while here we only consider n! when n is an integer, but its main results also apply to non-integer arguments. This note is longer and more elaborate than the previous ones, but the principal result is in the opening section. Only aficionados need go any further.