352 Dr. R, Key's Propositions containing 
to that series of propositions, I do not take upon me to decide. 
Certainly much of the matter, contained in the seven, is not 
new. But, so far as it is necessary to the principal design, its 
admission will be justified. And also new proof, of what is 
already known, is sometimes acceptable to the mathematician, 
and may be admitted as an advancement of science. On which 
ground the proof here given of the sixth proposition, and some 
other proofs, await the decision of the reader. 
An indirect result of the research, is merely geometrical 
and may be taken separately from perspective. This consists 
of the introductory propositions. On the one hand, it appeared 
necessary, or desirable, that these should accompany the per- 
spective propositions ; because of various references, in the 
latter, to the former. On the other hand, it seemed also 
desirable that the introductory propositions should be so far 
kept separate, as to enable any reader to peruse them, who 
would have declined all examination of their contents if 
interwoven with perspective. 
The properties of the circle, not improbably, lie open to 
abundant successful investigation. It is hoped that the intro- 
ductory propositions, of this paper, may have contributed a 
little to the stock of knowledge respecting that remarkable 
geometrical figure. 
To avoid crowding the plates with figures, I have, in a few 
instances, left it to the reader to construct his own figures for 
illustration, if they appear to be wanted. For instance, in cor. 6 
to introd. prop. X, and cor. i to introd. prop. XII. 
