39® Dr. R. Key's Propositions concerning 
be wholly out of LOR. In FM take FP towards X, and FQ 
the contrary way, each a mean proportional between FN and 
FO ; and draw PC and QK, perpendicular to PQ, cutting the 
directing line in C and K. Let the circle PG with the radius CP 
cut LR in G towards R, and let the circle QH with the radius 
KO cut it in H towards L. Let the M nearest to X be between 
X and P. In the directing line find the centre W of a circle 
passing through O and this M, cutting FM in U also, and LR 
in I beyond A. Then, for circles whose mean points are U 
and the M nearest to X, the directing point of the minor 
axis is I. If a circle move, and so that its mean point shall 
move in XM ; then, while this moves from that M to P, and 
on from P to U, the directing point of the minor axis moves 
accordingly from I to G, and back from G to I. While the 
mean point moves from U to F, from F to Q, then beyond Q 
without limit, such directing point, accordingly, moves from 
I to A, then from A to H, then approaches to A without limit. 
From these directing points triangles may be constructed*, 
and transferred to the plane of the picture ; giving lines 
parallel, as before, to the minor axes. Such, then, would have 
been a law observed, if the mean points had been in a right 
line. 
For, describe the circles PON, QON. Their centres aref 
C and K. Therefore they are the circles PG and OH ; that 
is, the circles OM J with mean points at P and O. Now, since 
all the circles OM have§ their centres in the directing line, and 
since, if a centre were assumed between C and K, the circum- 
ference through O could not meet XM, no centre is between 
* As LOA, GOA., &c; in prop. IX> X. 
I According to the construction ot prop. VI. 
j III Eucl. prop. 1, 37, 
S lb. 
