GEOMETRIC CONSTRUCTION. aed 
Thus to the points of the range on a tangent to K correspond 
the points of the range on a tangent to K’, corresponding 
tangents to K and K’ meeting on/. Hence the construction 
transforms the tangents to K into the tangents to K’ and de- 
termines a one-to-one correspondence between the two sets of 
tangents. 
Suppose that the point P moves along PQ and approaches 
A, its point of tangency with K; the construction shows that 
the corresponding point P’ will approach A’ the point of tan- 
gency of P’Q and K’. Hence in the limit the point of tan- 
gency of P’Q with K’ corresponds to the point of tangency of 
PQ with K. Therefore the construction determines a one-to- 
one correspondence between the points of K and the points of 
K’; the tangents at corresponding points meeting on J. 
Let P and P, be two points in z not on K, and let the tan- 
gents drawn from them to K meet / in Q and R, Q, and R,, 
respectively. Let ¢ be any tangent meeting these four fixed 
tangents in A, B, A,, B,. Since ¢ and / are both tangents to 
K, the cross-ratios (A BA,B,) and(QRQ,f,) are equal. The 
construction transforms P and P, into P’ andP,’; the tangents 
from these points to K into the tangents from P’ and P,’ to 
kK’; the tangent t into the tangent t’ and the points A, B, 
A,, B, on t into A’, B’, A,’, B,’, the points where ?¢’ cuts the 
tangents from P’ and P,/ to K’. Since ¢’ and / are tangents 
to K’, the cross-ratios (A’B’A,'B,’) and (QRQ,F,) are equal. 
Hence we have (ABA,B,)=(A'B’A,'B,'). Therefore the 
range of points on a tangent to K and the corresponding 
range of points on the corresponding tangent to K’ are pro- 
jectively related. 
We wish to show next that straight lines in 2 not tangent 
to K are transformed by our construction into straight lines 
ina’. Let P, bea point in x not on one of the tangents from 
P to K and let a pencil of lines be drawn in x through P, cut- 
ting PQ and PR in two ranges of points, which are therefore 
projectively related and in perspective position. Our con- 
struction transforms this configuration in z into the following 
