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 ...

38 8 REASON.
ciple the properties of the parabola or ellipse, but fiom thle
definitions of these curves? A similar observation may be extended to all the other theorems which the mathematician
demonstrates; and it is this observation (which, obvious as it
may seem, does not appear to have occurred, in all its force,
things exactly conformable to the definitions. There exist no points without magnitude; no lines without breadth, nor perfectly straight; no circles
with all their radii exactly equal, nor squares with all their angles perfectly
right." " To get rid of this difficulty, and, at the same time, to save the
credit of the supposed systems of necessary truth, it is customary to say
thlat the points, lines, circles, and squares which are the subject of geometry, exist in our conceptions merely, and are part of our minds; which
minds, by working on their own materials, construct an a priori science, the
ev idence of which is purely mental, and has nothing' whatever to do with
outward experience." This doctrine, however, lie maintains to be psychologicanl, incorrect; for " we cannot conceive a line without breadth; we can
form no mental picture of such a line; all the lines which we have in our
minds are lines possessing breadth." " A line, as defined by geometers, is
wholly inconceivable. We can reason about a line," lie admits, " as if it
had no breadth; " but this is only because we have a power of attendingc to
a part of our perception or conception, instead of the whole. The conclusion, therefore, in geometry, is Gnly so far an approxination to the truth, as
the points, lines, circles, etc., which are described in the definitions, are
approximations to the real lines, circles, etc., which actually exist.
"The peculiar accuracy," continues Mr. Mill, " supposed to be characteristic of the first principles of geometry, thus appears to be fictitious.
The assertions, on which the reasoninogs of the science are founded, do not,
any more than in other sciences, exactly correspond with the fact; but we
sup1pose that they do so, for the sake of tracing the consequences which follow from the supposition. The opinion of Dugald Stewart respecting the
foundations of geometry, is, I conceive, substantially correct;- that it is
built upon hypotheses; that it owes to this alone the peculiar certainty
supposed to dcistinguish it; and that, in any science whatever, by reasoning
from a set of hypotheses, we may obtain a body of conclusions as certain
as those of geometry;- that is, as strictly in accordance with the hypotheses, and as irresistibly compelling assent on cozndition that those hypotheses
are true. When, therefore, it is affirmed that the conclusions of geometry
are necessary truths, the necessity consists in reality only in this, that they
necessarily follow from the suppositions fiom which they are deduced.
These suppositions are so far from being necessary, that they are not even
true; they purposely depart, more or less widely, from the truth." —
Mill's Loqic, Amn. ed. pp. 148-151.