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SCIENCE 



[N. S. Vol. XLIII. No. 1101 



through it, then another point of the first 

 on this line, and so forth. It may be that 

 this process will close, the last line passing 

 through the first point. If it does close, 

 forming a polygon of n sides with vertices 

 on the first conic and sides tangent to the 

 second, then every point of the first is a 

 vertex of one such closed polygon, and 

 every tangent of the second is a side of one 

 such polygon of n sides. 



That is the first part of the theorem. 

 The second is this. Diagonals of all these 

 closed polygons, which omit the same num- 

 ber of consecutive vertices of the polygon, 

 are tangent to a fixed third conic; and the 

 dual statement is true concerning points of 

 intersection of non-consecutive sides. This 

 latter part of the theorem is true even if 

 the polygon is not closed. Prom some 

 points of view this scholium exceeds in im- 

 portance the principal theorem. 



These statements give us a specific atti- 

 tude toward the conies. We look upon the 

 first as a groove prepared to guide a set of 

 sliding points, and the second as a direc- 

 trix for lines joining the points. If the 

 lines are indefinitely extended, there will be 

 outlying systems of crossings; a first extra 

 set whose motion will describe a first extra 

 conic; a second extra set with its conic 

 locus, and so forth. The case where the 

 polygon is closed is that in which one of 

 these extra loci coincides with the first 

 conic. 



We may digress to notice a curious fact. 

 The sides of an inscribed hexagon meet in 

 15 points, namely, six on the conic, three 

 on the Pascal line and six which we may 

 term for the moment extra points. These 

 six extra points are vertices of a hexagon 

 circumscribed to a second conic. If now 

 the first hexagon, already inscribed to one 

 conic, becomes circumscriptible, then the 

 hexagon of the extra points, already cir- 

 cumscribed to a conic, becomes inscriptible 



to another. This separation of two prop- 

 erties which occur together in all polygons 

 of the Poncelet type is a situation deserving 

 further attention. 



To return to Poncelet: His discovery of 

 the mobility, or the infinite multitude, of 

 these polygons upon two fixed conies, pub- 

 lished in 1822, must have seemed to mathe- 

 maticians of that day as startling as the 

 announcement of a new genus of vertebrates 

 by a traveler returning from distant lands. 

 Its exact character had to be ascertained 

 and settled. The possibilities of variation 

 must be examined ; as, for example, whether 

 all the sides of the polygon need be required 

 to touch the same conic. Here it was seen 

 by Poncelet himself that if all conies con- 

 cerned pass through the same four basis 

 points, then it is sufficient for the purpose 

 if each side in its order touches its O'wn 

 assigned conic — all the vertices will stiU be 

 movable on their common track. After 

 this, it seems like a new proposition to 

 assert that the order in which the fixed 

 conies are touched by successive sides may 

 be varied, and still the polygon will close 

 in the same number of sides. And it is a 

 new proposition, as announced within the 

 last few years by Rohn, provided not 

 merely their order, but also their cyclic 

 order, is altered. Whether in this general- 

 ized figure the extra points still describe 

 loci of the same family, that I do not re- 

 member seeing demonstrated. 



The fertile mind of Jacobi seized the 

 germ idea of periodicity in this closed fig- 

 ure, so closely resembling sets of arguments 

 of the elliptic functions differing by aliquot 

 parts of a period. This suggestion was the 

 more natural because of the geometrical 

 diagrams current in the definition of ellip- 

 tic arguments. Only six years after the 

 date of Poncelet 's book, we find (1828) in 

 Crelle's Journal, Vol. III., Jacobi 's brief 

 and elegant essay on these polygons for the 



