The Poncelet-Steiner Theorem and Double Rulers

Abstract

In the early nineteenth century, Poncelet and Steiner dominated the revival of interest in pure geometry, as opposed to the methods of analytic geometry. Jacob Steiner (1796–1863) worked on his father’s farm until the age of nineteen, before going off to Berlin to become what some regard as the greatest geometer of modern times. Jean-Victor Poncelet (1788–1867) entered the French army corps of engineers just in time to take part in Napoleon’s disastrous 1812 campaign. After his capture by the Russians, Poncelet spent his time in a Moscow prison to good advantage, developing the concepts of projective geometry. Steiner also made significant contributions to this new method of geometric thinking. In 1822, Poncelet, inspired by the results of Mascheroni, gave indications of a proof that all the ruler and compass constructions could be carried out with the ruler alone, provided one circle with its center was given. Steiner published his detailed proof of this result in 1833. Both Poncelet and Steiner were ardent supporters of synthetic geometry and disliked analytic methods to the extent of attacking those who used them. Therefore, it is with apologies to both Poncelet and Steiner that we will use analytic geometry in proving the theorem that bears both their names. However, at this point, we would not consider doing it any other way.

It is a very different matter actually to carry out the constructions, i.e., with the instruments in hand, than to carry them through, if I may use the expression, simply by means of the tounge.

Steiner