By Daniel Pedoe

This revised variation of a mathematical vintage initially released in 1957 will carry to a brand new iteration of scholars the joy of investigating that easiest of mathematical figures, the circle. the writer has supplemented this new version with a different bankruptcy designed to introduce readers to the vocabulary of circle recommendations with which the readers of 2 generations in the past have been prevalent. Readers of Circles want merely be armed with paper, pencil, compass, and instantly area to discover nice excitement in following the structures and theorems. those that imagine that geometry utilizing Euclidean instruments died out with the traditional Greeks should be pleasantly shocked to benefit many attention-grabbing effects which have been in basic terms came across 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 extent in three-space, a version for non-Euclidean geometry, and the isoperimetric estate of the circle.

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1), §1. For the centre of W is (~), and if we suppose that tg lies in the plane Oxy of E3 , and represent 1W by the point P ( the orthogonal projection of P on to the plane Oxy is the centre of the circle W which is represented by the point P. 3. First properties of the representation The square of the radius of the circle le given by (1)is e2 + '2- _. Q=X 2 +y 2 -e=O. (l) As we saw in §2, this equation represents a paraboloid of revolution. This paraboloid Q plays a fundamental part in our investigation.

All of which touch Wand A', such that W, touches W62 and qWn, W2 touches W3 and W1 , ... , W,, touches '6n-1 and W6? Inverting W and W' into concentric circles, we see that the FIG. 26 problem does not always have a solution, but that when there is a solution, we can begin the chain by drawing any circle which touches W and W', and can then fit in the other members of the chain in an obvious manner. We conclude this discussion of coaxal circles by considering the radical axes, in pairs, of three circles W, '", by.

18 These points have already appeared in some of our proofs. We now define them. External centre of similitude. Let W and IF' be two circles of unequal radii, with centres C and C'. Let CP and C'P' be parallel radii drawn in the same sense. Then it is clear that P'P cuts U'C in a fixed point 0, which divides U'C externally in the ratio of the radii of the two circles. This is the external centre of similitude. If the ratio of the radii = k, then OP': OP = k. We also have OP. OQ = constant. OQ = constant.