By Cayley A.

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His most fascinating work, The Method, was rediscovered as recently as 1906. He may or may not have shouted ‘Eureka’ and run naked through the streets, but he was certainly murdered by a Roman soldier, an event that marks the end of an era in mathematics. 57 Archimedean spiral. Archimedes’ method Take a circle. Computing areas or perimeters of *inscribed and *circumscribed n-sided polygons generates two sequences, the first of which all underestimate the corresponding measurement of the circle, and the second sequence are all overestimates.

Apex (apices) See BASE (of a triangle) and PYRAMID. aphelion See APSE. apogee See APSE. Apollonius’ circle Given two points A and B in the plane and a constant k, the locus of all points P such that AP/PB = k is a circle. A circle obtained like this is an Apollonius’ circle. Taking k = 1 gives a straight line, so either this value must be excluded or, in this context, a straight line must be considered to be a special case of a circle. In the figure, k = 2. 52 • An interactive webpage where you can alter the ratio to see how the circle changes.

Additive inverse See INVERSE ELEMENT. adherent point A point of the *closure of a set. ad infinitum Repeating infinitely many times. adj Abbreviation for *adjoint. adjacency matrix For a *simple graph G, with n vertices v1, v2, …, vn, the adjacency matrix A is the n × n matrix [aij] 31 with aij = 1, if vi is joined to vj, and aij = 0, otherwise. The matrix A is *symmetric and the diagonal entries are zero. The number of ones in any row (or column) is equal to the *degree of the corresponding vertex.