Elements of the philosophy of the human mind. By Dugald Stewart. Rev. and abridged, with critical and explanatory notes, for the use of colleges and schools. By Francis Bowen ...

REASONING AND DEDUCTIVE EVIDENCE. 435
the same parallels, are equal; a theorem which appears, fivom a
very simple construction, to be only a few steps removed fiom
the fburth of the same book, in which the supposed application
of' the one triangle to the other, is the only medium of comparison from which their quality is inferred.
In general, it seems to be almost self-evident, that the equality
of two spaces can be demonstrated only by showing, either that
the one might be applied to the other, so that their boundaries
should exactly coincide; or that it is possible, by a geometrical
construction, to divide them into compartments, in such a manner,
that the sum of parts in the one may be proved to be equal to
the sum of'parts in the other, upon the principle of superposition.
To devise the easiest and simplest constructions for attaining
this end, is the object to which the skill and invention of the
geometer is chiefly directed.
Nor is it the geometer alone who reasons upon this principle.
If you wish to convince a person of plain understanding, who
is quite unacquainted with mathematics, of the truth of one of'
Euclic's theorems, it can only be done by exhibiling to his eye
operations exactly analogous to those which the geometer presents to the understanding. A good example of this occurs in
the sensible or experimental illustration which is sometimes
given of the forty-seventh proposition of Euclid's first book.
For this purpose, a card is cut into the form of a right angled
triangle, and square pieces of card are adapted to the different
sides; after whichi, by a simple and ingenious- contrivance, the
diffirent squares are so dissected, that those of the two sides
are made to cover the same space with the square of the
hypothenuse. In truth, this 2mode of comparison by a superposition, actual or ideal, is the only test of equtality which it is
possible to appeal to; and it is firom this, as seems friom a passage in Proclus to have been the opinion of Apollonius, that, in
point of logical rigor, the definition of geometrical equality
should have been taken.* The subject is discussed at great
* I do not think, however, that it would be fair, on this account, to censure E1aclld for the arrangement which he has adopted, as he has thereby