Febeuaky 4, 1916] 



SCIENCE 



151 



have given grounds for a hypothesis. But 

 Fuss limited his investigation, so Jaeobi 

 states, to the ease where the polygon is 

 symmetrical with respect to the common 

 diameter of the two circles. Symmetrical- 

 irregular polygons, he calls them ; and this 

 Fuss supposed to be essentially a restric- 

 tion upon the generality of the problem, 

 and hence he believed that he had solved 

 only under limitations the problem pro- 

 posed. This misapprehension apparently 

 persisted for 26 years, until the appear- 

 ance in 1822 of Poncelet 's memorable work : 

 "Traite des Proprietes Projectives des 

 Figures." Indeed there is indirect evi- 

 dence to this effect in an essay by Poncelet 

 himself, of the date 1817, in which he chal- 

 lenges his correspondent to solve the prob- 

 lem of inscribing in a given conic a polygon 

 of n sides, the sides to be tangent to a sec- 

 ond given conic. This problem as stated is, 

 as we now know, misleading, implying that 

 there is a solution, and that the number of 

 solutions is finite. Poncelet would hardly 

 have ventured to publish such a problem 

 had he not been sure that the mathematical 

 public of that day would accept it in good 

 faith. 



It would be quite certain also, even if we 

 had no direct knowledge of the fact other- 

 wise, that the relations of coUinearity and 

 correlativity or reciprocity with respect to 

 a conic were not at all commonly under- 

 stood prior to 1822. The employment of 

 transformations to derive one solution of a 

 problem from another was not yet a recog- 

 nized preliminary to all discussion. The 

 student of conies to-day mil reflect at once 

 that two conies not specialized in situation 

 have one self -polar triangle in common, and 

 are transformed into themselves by three 

 collineations or projectivities besides the 

 identity, and are transformed simultane- 

 ously into each other by four reciprocities 

 or polarities with respect to a third fixed 



conic. Thus to-day we should see in advance 

 that any one triangle, or one pentagon, in- 

 scribed in one conic and circumscribed to 

 another, implies seven others of the same 

 sort. Solutions of Poncelet 's problem must 

 occur at least in sets of eight ; but this fact, 

 apparent from Poncelet 's own discoveries, 

 appears to have escaped his attention, and 

 still less was it present to the minds of his 

 contemporaries. 



Knowledge of the investigations of Fuss 

 and of Euler would have been almost use- 

 less to Poncelet. For the far superior gen- 

 erality of his problem, that of two conies in 

 place of two circles, his method of projec- 

 tion is responsible. This allowed him to 

 use metric properties of circles and draw 

 conclusions concerning any two curves of 

 the second order. But the discovery of his 

 famous theorem on polygons was nothing 

 less than a stroke of genius. Many have 

 been quoted as authors of the saying that 

 invention or discovery is the principal thing 

 in geometry, while the proof is a relatively 

 easy matter. In this case, however, the 

 proof also is ingenious, carried on by the 

 exclusively synthetic method. But the per- 

 ception of the theorem, preceding its proof, 

 escapes explanation from anything that had 

 gone before. Were that his only contribu- 

 tion to our knowledge of geometry, it would 

 ensure him grateful recognition from later 

 students — as the compeer of Apollonius 

 who gave us the foci of a conic, Desargues 

 who first perceived poles and polars, New- 

 ton who described the organic construction 

 of conies, and the immortal Pascal with his 

 hexagon. Let us rehearse the theorem 

 which gives a generic name to Poncelet 

 polygons. 



Of two given conies, call one the first and 

 consider its points; call the other the sec- 

 ond and consider its tangents. Form a 

 broken line by taking a point of the first 

 curve, a line of the second that passes 



