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SCIENCE 



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



mentary to, not extracted from, tlie mate- 

 rial world. Knowable they are, therefore, 

 by their very constitution. But who can 

 ever conceive of them as limited in number ? 

 Who can imagine that ever in the future it 

 could come to pass that there should be no 

 more geometric concepts to be investigated ; 

 that a point might be attained where the 

 mind of the mathematician should rest 

 satisfied, all its curiosity appeased? 



Connected with this contrast in the 

 source of its objects is the slowness with 

 which new objects in geometry emerge and 

 diffuse into general knowledge. Called into 

 being by shifting stimuli, multitudes of new 

 systems of relations are invented and named 

 and investigated; but most of them are 

 speedily forgotten (or perhaps only dimly 

 apprehended even by the discoverer), and 

 very few in a century are those which sur- 

 vive to become the valued heritage of later 

 generations. 



There are many occasions when we meet 

 to discuss only what is new. The present,' 

 however, is a fitting occasion for reviewing 

 together some of the treasures handed on to 

 us by geometricians of the past, and for 

 stimulating our own ardor by the rehearsal 

 of the fortunes and successes of earlier 

 workers in our part of the field of science. 

 The polygons of Poncelet were new a hun- 

 dred years ago, and are not yet forgotten, 

 but seem rather to attract increasingly the 

 interest and attention of geometricians. I 

 invite you to enjoy with me, since though 

 not unknown they are not yet in the class 

 of familiar objects, a rapid survey of their 

 character and development. 



For many centuries before Euler stu- 

 dents of geometry had found interest in 

 circles inscribed in a triangle and circum- 

 scribed to it. Usually their centers do not 

 coincide. One circle may be kept station- 

 ary, while the triangle varies, and with it 

 vary also the center of the other circle and 



its radius. Euler may have been the first 

 to write out the relation that connects these 

 three quantities, the two radii and the dis- 

 tance of the centers : B" = 2Rr + <P, or it 

 may have been discussed a hundred times 

 before. Publication of this relation led to 

 the study of analogous relations for poly- 

 gons of more sides, Fuss in St. Petersburg, 

 and some years later Steiner in Berlin, 

 carrying the problem farthest, finding re- 

 sults for polygons of 4, 5, 6, 7 and of 8 

 sides. The case of regular polygons, for 

 which the inscribed and circumscribed 

 circles are concentric (c^ = 0) is elemen- 

 tary, and will always stimulate interest in 

 the more general problem. 



While attention was directed to finding 

 an algebraic relation corresponding to 

 a given geometric diagram, for a long time 

 no one seems to have inquired whether this 

 relation was merely a necessary condition, 

 or whether it might also be a sufficient con- 

 dition for the construction of the diagram. 

 If two circles are drawn, satisfying the 

 condition for a triangle: R- = 2Br-\- d^, 

 can one always determine the triangle in- 

 scribed in the circle radius R and having 

 its sides all tangent to the circle of radius 

 r? And is there only one such triangle in 

 each case, or some finite number greater 

 than one ? What of the ease where the tri- 

 angle (or polygon of 4, 5 or more sides) is 

 regular — is it exceptional that for that ease 

 there are an infinite number of polygons 

 which satisfy the requirements, provided 

 there is one such ? 



It is not easy to apprehend the state of 

 geometric knowledge in 1796, when Fuss 

 wrote on this subject. He certainly sup- 

 posed that a triangle could occur singly, 

 and was unaware that others can always be 

 inscribed and circumscribed to the same 

 pair of circles. It would seem as though the 

 roughest kind of experimentation would 

 have shown the truth, or at least would 



