This paper contains (i) a short history of the geometrical theorem proposed in 1840 by Prof. Lehmus of Berlin to Jacob Steiner—“If BJY, CJZ are equal bisectors of the base angles of a triangle ABC, then AB equals AC,” (ii) a selection of some half-dozen solutions from the 50 or 60 that have been given, (iii) some discussion of the logical points raised, and (iv) a list of references to the extensive literature of the subject.

An object to which we were all introduced at an early stage in mechanics is the inextensible string. This appears frequently without causing much trouble, but there is one type of problem which, in my opinion, stands apart from the rest, and which certainly caused me a lot of trouble. Such a problem is when impulses are given to a system which includes an inextensible string, as, for example, a system consisting of two rigid parts joined by a string. If an impulse is applied to one of these parts, an impulsive tension (T) may be set up in the string, which, in turn, gives an impulse to the other part. One new quantity, T, has appeared, and one equation in addition to the ordinary dynamical equations is thus required before the problem of finding the change in motion of the system can be solved. It is at this stage that opinions can differ, for this extra equation depends essentially upon what concept of an inextensible string is being adopted, and there is more than one. The usual procedure is to employ a “geometrical equation” based upon the argument that the two ends must have equal component velocities in the line of the string as long as the string is taut. This seems almost obvious when described in such general terms, and is followed by such eminent writers as Routh and Loney, amongst others. I suggest, however, that this method implies a concept which leads to results contrary to common-sense and to everyday experience. This is best illustrated by a simple example.

Numerous proofs have been given of this familiar theorem, which states that if a1, a2, …, an are positive, and not all equal, then

The following is an elementary proof by induction, which I have not seen used before. It is, of course, not claimed to be novel, and not likely to be so.

Tullio Levi-Civita, Honorary Member of the Edinburgh Mathematical Society, was born at Padua on March 29th, 1873, and died in Rome on December 29th, 1941.