R.K. Rubugunday Distinguished Lecture:
On a Saraband theme of circles and some variations
Chennai Mathematical Institute.
The straight edge and a pair of compasses were the instruments used from ancient times in the West to do geometrical constructions in the plane. When the Italian mathematician Lorenzo Mascheroni(1750-1800 A.D) met Napoleon Bonaparte (during the latter's conquest of Northern Italy) to tell him of his discovery that one could dispense with the straight edge altogether and achieve all the geometrical constructions with a pair of compasses alone, Napoleon (being interested in Mathematics as he was) was deeply impressed and discussed this, when he came back to France with the French mathematicians Laplace and Lagrange, who in turn were surprised by this result. As it happens often, Mascheroni's result had been anticipated more than a century back by an unknown Danish mathematician Georg.Mohr and this fact was discovered by accident in 1928 by a student of the Danish Geometer Hjelmslev, in an old bookshop in Copenhagen, when he was browsing through a book on Euclidean geometry written by Mohr in 1672.
The Swiss geometer J.Steiner (1796-1863), inspired by the French geometer Victor Poncelet proved on the other hand that the compasses can be discarded and all the geometrical constructions in the plane could be done only with a straight edge, provided one is given in advance a circle in the plane with its centre marked. Hilbert, who was interested in axiomatizing Euclidean geometry, proved in his lectures that the condition of Steiner about the centre of the given circle being given is indeed essential, by showing that the centre of a given circle in the plane cannot be determined by the straight edge alone. Acting on Hilbert's suggestion, one of his students, D.Cauer proved more generally in 1913 (Math.Annalen) that even if one starts with two non-intersecting circles in the plane, it is not possible to find their centres by straight edge alone.On the other hand, he showed that it is possible to find the centres of two given circles with straightedge alone if they intersect, touch or are concentric. We shall sketch proofs of these results of Caur.
Circles have played an important role in the development of Mathematics. To quote but two more examples, one could first cite the beautiful theory of Jacobi(Crelle Bd3) which defines elliptic integrals associated to a pair of circles one inside the other and using them for instance to prove the celebrated theorem of Poncelet on the existence of polygons inscribed in a circle and circumscribing the other. One could also mention the concept of ``Ford circles'' and their relation ship with the geometry of the upper half plane and the ``modular Group''.
Let me end with the sobering thought that the great Johamnn Sebastian Bach composed his ``Goldberg variations'' as a soporific for Count Keyserling, the Russian envoy at the Dresden court.