some Properties of Tangents to Circles; ^c. 351 
demonstration puts an end to further inquiry about the point 
to which they might be imagined to converge, and wholly 
about such convergency in them ; and leaves us free to 
inquire after any other law or regularity whicli the axes may 
observe. 
Now the object of the lemma (with its first corollary), 
prefixed to the eightlt of the propositions on perspective, is to 
prove that lines, in a picture, cannot converge to a point ; 
unless the original lines which they represent happen so to 
converge or else to be parallel. The inference is, that the 
axes cf the ellipses cannot so converge, unless the chords 
Vvhich they represent should so converge or be parallel : a 
case which may rarely happen, and, if happening, may not be 
readily discernible. 
The convergency to a point, by the axes, being therefore 
dismissed, the four remaining propositions, after the lemma, 
exhibit nevertheless a regularity in the directions of the axes. 
Though these do not converge to a point, they are (if the rea- 
soning be just) parallel respectively to other lines which do 
converge to a point. To find each of these other lines separately 
and accurately, is not the object proposed. But certain limits 
are pointed out,' within which the artist is to k?ep; and a 
regularity marked, by which the variation in the directions of 
successive axes is to be guided: wdiich, it is hoped, will pro- 
duce a sufficient accuracy in a great number of instances. 
And, where any one or more circles require singly a strict 
accuracy, to this the construction of the sixth proposition is 
adapted. 
How much, or whether any part, of the first seven propo- 
sitions on perspective, might iiave been omitted, without injury 
Z z 2 
