theory indicated to practice where it should be sought for, 

 and there it was found. It may not be unacceptable to the 

 Society if I give — though it is a digression from the main 

 subject of this paper — an outline or sketch of the process 

 by which the existence of this planet was first suspected 

 and finally ascertained. It is known, I presume, to all 

 here, that the orbits of the planets are ellipses nearly in one 

 plane, having the sun in then common focus. W hy was not 

 this planet discovered sooner ? it may be asked. It is, in fact, 

 as Lord Bacon said of another discovery — ' pottos temporis 

 epiam partus ingenii' — the birth rather of time than of genius. 

 The discovery of the planet Herschell was a necessary preli- 

 minary to the discovery of Neptune. Herschell was the indi- 

 cator or the gage which led to the discovery of the new 

 planet. It will be obvious, from considering the enormous 

 mass of the sun, and the close proximity to it of all the older 

 planets, that Neptune coidd not have been anticipated, how- 

 ever perfect the theory of the solar system, had not Herschell 

 been first discovered ; and centuries hence, a perturbation in 

 the motions of Neptune may indicate the existence of a still 

 more distant sentinel of our system, which no human eye may 

 yet discover. 



" But to return to the more immediate subject of this 

 paper — the duality of geometrical relations. Until the begin- 

 ning of the present century, the propositions of geometry were 

 deduced by direct investigation, one from the other, and the 

 existence, on the discovery of any one property of extension, 

 did not lead the inquirer further; but about this time, 31. 

 Poncelet, an officer of engineers in the army which invaded 

 Russia under Buonaparte, having, in the course of that disas- 

 trous campaign, been taken a prisoner of war, in one of the 

 prisons of that country, with only a few mathematical books 

 his companions, made the remarkable discovery that the pro- 

 positions of geometry arc dual ; that is to say, any proposition 

 of geometry, any property of lines, circles, triangles, or conic 



