By Daniel Pedoe

This revised version of a mathematical vintage initially released in 1957 will carry to a brand new iteration of scholars the joy of investigating that least difficult of mathematical figures, the circle. the writer has supplemented this new version with a unique bankruptcy designed to introduce readers to the vocabulary of circle options with which the readers of 2 generations in the past have been general. Readers of Circles desire merely be armed with paper, pencil, compass, and directly side to discover nice excitement in following the buildings and theorems. those that imagine that geometry utilizing Euclidean instruments died out with the traditional Greeks may be pleasantly stunned to profit many attention-grabbing effects that have been in simple terms came upon nowa days. beginners and specialists alike will locate a lot to enlighten them in chapters facing the illustration of a circle by way of some degree in three-space, a version for non-Euclidean geometry, and the isoperimetric estate of the circle.

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Inversion This is a one-to-one transformation of the points of the plane by FIG. 6 means of a given circle, which we shall call X, of radius k and centre 0. To obtain the transform X' of any given point X, or the inverse of X in the circle X, we join X to 0, and find X' on OX such that OX. OX' = k2. The point 0 itself is excluded from the points of the plane which may be transformed. It is clear that (i) X is the inverse of X'; (ii) points on £ transform into themselves; (iii) if A, A' are the ends of the diameter of E through X, then X, X' are harmonic conjugates with respect to A, A'.

XD, so that X is on the radical axis of W, and W2. We also note that if the contacts at A and B are of like or unlike type according as those at C and D are, then the contacts at A and C are of like or unlike type according as those at B and D are. 9r2. * 2 contains a centre of similitude of WI' and 2. We now return to our problem, and assume, for simplicity, that the radii of W,, W2' S3 are three distinct numbers, so that all the oentres of similitude exist. Let SO be a circle touching the three circles externally.

7 As a corollary to this remark, we see that if % is any circle orthogonal to X, all points on W invert into points on W, for OX. OX' = k2. It is sometimes necessary to consider the process of inversion when X, the circle of inversion, is a line. If we keep A fixed, and let A' move off to infinity, we see that X' moves towards the geometrical image of X in the resulting line. -I . X' l | A - | X z FIG. 8 For future application we now give a construction, using only a -X~ FIG. 2 pair of compasses, by which we can find the inverse of a point X in a circle 2 of given centre 0.