School exercise books have usually a transitory existence. Even if we ourselves retain them as souvenirs of our schooldays—and we seldom do—then our descendants have no such sentimental regard for them, and consign them to the waste-paper basket when we depart this life. A school exercise book of 130 years ago is therefore remarkable for its longevity, as well as interesting for the light it throws on the teachings of that time.

One of the pleasant aspects of research into mathematical history is the way in which existing material, passed over in the conventional account, may allow us not only to establish the bare, if unexpected, fact of a priority but, more importantly, to assess the significance of major currents in mathematical thought with greater precision. Briggs’ partial discovery, in anticipation of Newton, of the general binomial expansion,

is an excellent case in point.

Proofs of the prime number theorem are extremely hard to follow, and leave the impression, at least among amateurs, that the essential property of prime numbers, namely their primeness, plays very little part in the argument.

Simple reasoning, based on the rules of probability, can however give a very fair indication of the way in which the density of primes in the region x varies with x.

In the preceding note Mr de Visme writes y’(x) for the “density” of primes in the neighbourhood of x and proves heuristically that

1. Introduction. If F and G are differentiate functions of one variable, the “function of a function” rule is

where f and g denote the derivatives of F and G respectively.

1. The preceding paper considers the most general theorem on change of variable in a Riemann integral: If g(t) is integrable over [a, b] and f (x) is integrable over G ([a, b]), then (G(t))g(t) dt exists and equals (x)dx.

These designs were prompted by the “pursuit curves” of Mr. I. J. Good The first one consists of his six ‘hyperboloids’ and the second an equal number of ‘spearheads’ The former possesses what may be termed cyclical symmetry and the latter triple axial symmetry

There seems to be a technical improvement in these designs over those of Mr. Good’s insofar as the black areas at the centre of his triangular doodles are conspicuous by their absence over here.

The representation of a twisted cubic as the intersection of a hyperboloid and cylinder with common generator makes an interesting model. This note gives details of its construction.

Let w be a point and ABCDEF a polygon, convex or not, which we call (P). Projecting w orthogonally on the sides AB, BC, CD, DE, EF, FA of (P), and joining the respective projections, we obtain the polygon A1B1C1D1E1F1, called (P1) [see Figure 1]. We call (P1) the first pedal of w with respect to (P).