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 ...

L436 i'REASONINrG AND DEDUCTIVE EYITDENCE.
length and with much acuteness, as well as' learning, in one of
the mathematical lectures of Dr. Barrow; to which I must
refer those readers who may wish to see it more fully illustrated.
%lentit y# atnd equalfity are not convertible terms. - I am strongly
inclined to suspect, that most of the writers who have maintained that all mathematical evidence resolves ultimately into
the perception of identity, have had a secret reference in their
own minds to the doctrine just stated; and that they have imposed on themselves, by using the words identity and equality
as literally synonymous and convertible terms. This does not
seem to be at all consistent, either in point of expression or of
fhct, with sound logic. When it is affirmed, for instance, that
"6if two straight lines in a circle intersect each other, the rectangle contained by the segments of the one is equal to the
rectangle contained by the segments of the other;" can it with
any propriety be said, that the relation between these rectangles
may be expressed by the formula a - a? Or, to take a case yet
most ingeniously and dexterously tried to keep out of the view of the
student some very puzzling questions, to which it is not possible to give a
satisfactory answer till a considerable progress has been made in the elements. When it is stated in the form of a self-evident truth, that magnitudes which coincide, or which exactly fill the same space, are equal to one
another, the beginner readily yields his assent to the proposition; and
this assent, without going any further, is all that is required in any of the
demonstrations of the first six books; whereas, if the proposition were
converted into a definition, by saying, "Equal magnitudes are those which
coincide, or which exactly fill the same space;" the question would immediately occur, Are no magnitudes equal but those to which this test of
equality can be applied. Can the relation of equality not subsist between
magnitudes which differ from each other in figure? In reply to this question, it would be necessary to explain the definition, by adding, That those
magnitudes likewise are said to be equal, which are capable of being
divided or dissected in such a manner, that the parts of the one may severally coincide with the parts of the other; a conception much too refined
and complicated for the generality of students at their first outset; and
which, if it were fully and clearly apprehended, would plunge them at once
into the profound speculation concerning the comparison of rectilinear
with?urvilinear figures,.