Dnieper and incarcerated at Saratoff, Poncelet employed the leisure captivity left him in the demonstration of the principles which he has developed in the 'Traité des propriétés projectives des figures' issued in 1822, and in the great memoirs on reciprocal polars and on harmonic means, which go back nearly to the same epoch. So we may say the modern geometry was born at Saratoff.
Renewing the chain broken since Pascal and Desargues, Poncelet introduced at the same time homology and reciprocal polars, putting thus in evidence, from the beginning, the fruitful ideas on which the science has evolved during fifty years.
Presented in opposition to analytic geometry, the methods of Poncelet were not favorably received by the French analysts. But such were their importance and their novelty, that without delay they aroused, from divers sides, the most profound researches.
Poncelet had been alone in discovering the principles; on the contrary, many geometers appeared almost simultaneously to study them on all sides and to deduce from them the essential results which they implicitly contained.
At this epoch, Gergonne was brilliantly editing a periodical which has to-day for the history of geometry an inestimable value. The Annales de Mathématiques, published at Nimes from 1810 to 1831. was during more than fifteen years the only journal in the entire world devoted exclusively to mathematical researches.
Gergonne, who, in many regards, was a model editor for a scientific journal, had the defects of his qualities; he collaborated, often against their will, with the authors of the memoirs sent him, rewrote them, and sometimes made them say more or less than they would have wished. Be that as it may, he was greatly struck by the originality and range of Poncelet's discoveries.
In geometry some simple methods of transformation of figures were already known; homology even had been employed in the plane, but without extending it to space, as did Poncelet, and especially without recognizing its power and fruitfulness. Moreover all these transformations were punctual, that is to say they made correspond a point to a point.
In introducing polar reciprocals, Poncelet was in the highest degree creative, because he gave the first example of a transformation in which to a point corresponded something other than a point.
Every method of transformation enables its to multiply the number of theorems, but that of polar reciprocals had the advantage of making correspond to a proposition another proposition of wholly different aspect. This was a fact essentially new. To put it in evidence, Gergonne invented the system, which since has had so much success, of memoirs printed in double columns with correlative
