If L, M, N are the feet of the perpendiculars from P to the sides of a triangle ABC, the circle through L, M, N is the pedal circle of P with respect to the triangle ABC.

Is it always possible to determine a real number to an assigned number of decimal places? Can we always tell how many terms of a convergent sequence must be taken to determine its limit to, say, four decimal places? How much information does the convergence of a sequence give us about the decimal expansion of the limit? These are the questions we shall consider; we shall find that the answer to the first two is “no”. To the third question the answer is that convergence itself gives very little information but that “most” convergent sequences have in fact a subtler property which gives precise information about the expansion of the limit, the rate at which the successive digits of the expansion can be determined and the number of times a digit can recur. The questions we have raised are of practical importance for it is with the approximations-to-so-many-places, not with real numbers themselves, that all practical mathematics is concerned.

Preliminary note. Beginners find two ways of regarding an ellipse easy to understand. The first presents it as the curve traced out by a point P (Fig. 1) when r + r′, the sum of P’s distances from two fixed points F and F′, is a constant, 2a. From this definition come the equivalences AA′ = 2a, CA = CA′ = a, and CF = CF′ = ae, where e is a fraction less than unity C is the “centre” of the ellipse, F and F′ its “foci”, CA = a the “semi-axis major”, CB = b the “semi-axis minor”, and e the “eccentricity”

Problems about the American city road-system with its streets running east and west and its avenues running north and south have appeared from time to time in examination papers and are interesting because they give relations between binomial coefficients which are harder to get by straightforward algebra. After illustrating some of the methods by which difficult-looking results can be obtained simply, I shall obtain a simple-looking result with difficulty and ask for an easier way of getting it.

Definition: ABC is a triangle; A′B′C′ is its medial triangle; L, M, N are the feet of the perpendiculars from A, B, C respectively on a straight line, σ1 σ2, σ3 are three circles having their centres at A′, B′, C′ respectively such that σ1 passes through M and N, σ2 through N and L, and σ3 through L and M Then it is readily seen that the radical axes of the three circles taken in pairs meet in a point W—the radical centre of the circles. Again, since L is a common point of σ2 and σ3, their radical axis is the line through L at right angles to their line of centres B′C′, and hence to BC. Hence the lines through L, M, N at right angles to BC, CA, AB respectively meet at W, which is said to be the orthopole of the line LMN The common radical circle of σ1, σ2, σ3 is called the Orthopolar circle of the line LMN. Let us consider a few applications of the properties of this circle and its centre.