l ap Mr. Woodhouse on the Independence of the 
of the sign to the thing signified, is supposed to be the chief 
cause of the superior clearness of geometrical demonstration.* 
Another cause may perhaps be thought to exist in this circum- 
stance, that whatever is demonstrated, of a triangle or other 
diagram, considered as the representative of all triangles and 
diagrams, is moreover demonstrated of that individual triangle 
or diagram. A third, and more satisfactory cause than the last, 
may be, that in investigation, for the purpose of preventing 
ambiguity and mistake, it is frequently necessary to recur from 
the sign to the thing signified ; which is more easily done, the 
less general and arbitrary the modes of representation are ; and, 
consequently, in geometry more easily than in algebra. 
I do not pretend to have assigned, accurately, and all, the 
causes of perspicuity of geometrical reasoning. It may depend 
on certain intellectual acts and processes, which it is beyond 
the power of philosophy to explain. The circumstance, how- 
ever, of the signs employed in geometry being natural signs, 
will prove its perspicuity only to a certain extent, and in certain 
cases. It must fail to prove it, when the properties of solids are 
treated geometrically ; because the representation of solids on a 
plane by diagrams, is not a natural representation, that is, would 
not suggest to all minds the same tangible portion of extension. 
It must fail likewise to prove it, in questions concerning radii 
of curvature, areas of curves, &c. or in all questions in which 
the fluxionary or differential Calculus is usually employed. The 
* Does there not, however, here arise a consideration that takes away from the 
cause of the perspicuity of geometrical demonstration ? For the reasoning with a 
diagram cannot be generally true, except the diagram be considered abstractedly, and 
independent of those peculiar and distinguishing properties that determine its indi- 
viduality./ 
