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 ...

4-32 REASONIN G- AND DEDUCTIVE EVIDUENCE.
ples which are essentially of the same description, that all the
more complicated results in the science are derived.
Tihe problems of geometry are as hytpothet/cal as its theoremEs.
— To this general conclusion, with respect to the nature of
mathematical demonstration,. an exception may perhaps be, at
first sight, apprehended to occur, in our reasonings concerning
geometrical problems; all of these reasonings, as is well known,
resting ultimately upon a particular class of' principles called
postulates, which are commonly understood to be so very nearly
akin to axioms, that both might, without impropriety, be comprehended under the same name. "The definition of a postulate," says the learned and ingenious Dr. Hutton,' "will nearly
agree also to an axiom, which is a self-evident theorem, as a
postulate is a self-evident problem." The same author, in
another part of his work, quotes a remark from Dr. Barrow,
that "there is the same affinity between postulates and problems,
as between axioms and theorems."
In opposition to these very high authorities, I have no hesitation to assert, that it is with the def)ition.s of Euclid, and not
wvith the axiorms, that the postulates ought to be compared, in
respect of their logical character and importance;- inasmuch
as all the demonstrations in plane geometry are ultimately
founded on the former, and all the constructions which it recognizes as legitimate, may be resolved ultimately into the latter.
To this remark it may be added, that, according to Euclid's
view of the subject, the problems of geometry are not less hypothetical and speculative, (or, to adopt the phraseology of some
late writers, not less objects of pure reason,) than the theorems;
the possibility of drawing a mathematical straight line, and of
describing a mathematical circle, being assumed in the construction of every problem, in a way quite analogous to that in
which the enunciation of a theorem assumes the existence of
straight lines, and of circles corresponding to their mathematical
definitions. The reasoning, therefore, on which the solution of a
problem rests, is not less demonstrative than that which is employed in proof of a theorem. Grant the possibility of the three
operations described in the postulates, and the correctness of