26o ROCHESTER ACADEMY OF SCIENCE. [April 9, 



In fig. 2, I supposed the circle T to enlarge toward the left, the 

 dexter vertex remaining fixed. 1 shall now let the circle diminish, 

 the dexter vertex being always fixed. The circle 1 will also diminish 

 as shown in fig. 15. This diminution will continue until both become 

 points at the same moment. Then as the sinister vertex of T passes 

 through the dexter vertex to the right, I begins to enlarge to the left, 

 and finally T and / coincide, as shown in fig. 16, and we have the 

 case of the geometries : — The product of the segments of a pivoted 

 secant is constant. 



So far I have spoken only of what might be called positive 

 inversion, that is, inversion where the distance PI is laid off along 

 the line FT. If the distance PI be laid off backward instead 

 of along PI, we get what might be called negative inversion, which 

 differs from the case we have been considering in that the / circle 

 appears on the left of the curve of inversion in an exactly symmetrical 

 position to that for positive inversion. 



Thus figs. 15 and 16 would become figs. 17 and 20. Figs. 18 and 

 19 show the intermediate steps. 



Fig. 20 brings us to another familiar theorem of Geometry, viz.: 

 If any chord is drawn through a fixed point within a circle, the 

 product of its segments is constant. 



You will notice the close connection between these two elemen- 

 tary theorems of Geometry, a connection which the geometries fail 

 to bring out, but which could be very plainly shown, even in elemen- 

 tary works, if the two theorems were combined into one, as follows : — 



The product of the segments of a pivoted secant made by the 

 pivot (internal or external) is equal to the square of the line from the 

 pivot to the circle, the line being perpendicular to a diameter through 

 its extremity. If the pivot is internal the diameter will pass through 

 it ; if external the diameter will pass through the end of the line 

 which is on the circle. 



For the case in which / is a straight line, those interested in 

 mathematics will recognize a special case oi pole and polar, for which 

 many interesting correspondences might be drawn. 



Hon. Martin W. Cooke discussed and illustrated with diagrams 

 the law of the ellipse described by a body thrown from a revolving 

 planet. 



