Since a link in a plane has three degrees of freedom the simplest closed plane mechanism with constrained motion must have four lower pairs, i.e. one more than the number of degrees of freedom. If the lower pairs are turning pairs, the mechanism is the well-known four-bar linkage. One or two turning pairs may be replaced by sliding pairs without affecting constrained motion, but not less than two turning pairs must remain in a closed link group.

Every student of calculus learns that one typically solves a differential equation by integrating it. However, as Euler showed in his 1758 paper (E236), Exposition de quelques paradoxes dans le calcul intégral (Explanation of certain paradoxes in integral calculus) [1], there are differential equations that can be solved by actually differentiating them again. This initially seems paradoxical or, as Euler describes it in the introduction of his paper:

Here I intend to explain a paradox in integral calculus that will seem rather strange: this is that we sometimes encounter differential equations in which it would seem very difficult to find the integrals by the rules of integral calculus yet are still easily found. not by the method of integration. but rather in differentiating the proposed equation again; so in these cases, a repeated differentiation leads us to the sought integral.

This paper is about the classification of frieze groups. A frieze is a decorative strip of paper (or wood, stone,…) on which a pattern is produced by the periodic repetition of a picture along the strip. A symmetry of a frieze is an isometry of the plane that leaves the pattern unchanged, and a frieze group is the group of symmetries of some frieze. A popular exercise is to start with a given frieze and then try to identify its frieze group. However, in order to do this we need to know that there are only seven possible frieze groups, and what these groups are. Seven friezes, with different frieze groups, are illustrated below, in such a way that each pattern is invariant under the same translation, namely x → x + 1. Our task is to show that (up to a change in the motif, and a simple change of coordinates in the plane) these are the only frieze patterns.

Pantactic squares were invented and studied by Astle [1]. From a review by Holzer [2] we learned that the number of distinct pantactic squares remained unknown; Astle himself only mentions that the number of distinguishable pantactic squares runs into hundreds. Therefore we turned the problem over to the computer. By a simple programme in FORTRAN, run on our CDC 3600/3200 computer, we found altogether 800 distinguishable pantactic squares.

It follows at once from Pythagoras’ theorem about a right-angled triangle that the sum of the squares of the lengths of the diagonals of a rectangle is equal to the sum of the squares of the lengths of the four sides. Apollonius showed that the assertion holds for a parallelogram and, more recently, Amir-Moez and Hamilton gave a generalization to quadrilaterals by introducing a correction term which depends on the distance between the mid-points of the diagonals.

Many members of the Association know the Gazette only in its hampered and restricted war-time form. The time is therefore opportune for some remarks on the pre-war Gazette and on what may reasonably be expected from the post-war Gazette. The history of the Gazette, however, cannot be fully understood without some reference to the history of the Association; such reference need be only brief, since a full account is available in the admirable Presidential Address by Mr. A. W Siddons (XX, No. 237, February, 1936).

“We are all pacifists now”; or if not, it is difficult to see what excuse we have for living in the modern world. I feel therefore that a word of apology is needed for a paper on a military subject. But I hope you will acquit me of any intention to revive the hatred and bloodthirsty panic that poisoned the years 1914 to 1918, because the activity that I have to tell you about is not concerned with the destruction of life but rather with the saving of it.

The entertaining book, There’s one born every minute, by Harry Blackstone, presents many situations where most people would intuitively be prepared to wager even money on the outcome, whereas a proper study of the probabilities concerned shows that the bet is heavily biased. One particular problem is interesting because the author claims to have sought evaluation of the probability concerned, without success. It is as follows: