OF MATHEMATICAL REASONING.
1. _Discursive Reasoning._--WE have thus seen that our
notions of space, time, and their modifications, necessarily
involve a certain activity of the mind; and that the
conditions of this activity form the foundations of those
sciences which have the relations of space, time, and
number, for their object. Upon the fundamental principles
thus established, the various sciences which are included in
the term _Pure Mathematics_, (Geometry, Algebra,
Trigonometry, Conic Sections, and the rest of the Higher
Geometry, the Differential Calculus, and the like,) are
built up by a series of reasonings. These reasonings are
subject to the rules of Logic, as we have already remarked;
nor is it necessary here to dwell long on the nature and
rules of such processes. But we may here notice that such
processes are termed _discursive_, in opposition to the
operations by which we acquire our fundamental principles,
which are, as we have seen, _intuitive_. This opposition was
formerly very familiar to our writers; as Milton,--
. . . Thus the soul reason receives,
Discursive or intuitive.--_Paradise Lost_, v. 438.
For in such reasonings we obtain our conclusions, not by
looking at our conceptions steadily in one view, which is
_intuition_, but by passing from one view to another, like
those who run from place to place (_discursus_). Thus a
straight line may be at the same time a side of a triangle
and a radius of a circle: and in the first proposition of
Euclid a line is considered, first in one of these
relations, and then in the other, and thus the sides of a
certain triangle are proved to be equal. And by this
'discourse of reason,' as by our older {148} writers it was
termed, we set forth from those axioms which we perceive by
intuition, travel securely over a vast and varied region,
and become possessed of a copious store of mathematical
truths.
2. _Technical Terms of Reasoning._--The reasoning of
mathematics, thus proceeding from a few simple principles to
many truths, is conducted according to the rules of Logic.
If it be necessary, mathematical proofs may be reduced to
logical forms, and expressed in Syllogisms, consisting of
major, minor, and conclusion. But in most cases the
syllogism is of that kind which is called by logical writers
an _Enthymeme_; a word which implies something existing in
the thoughts only, and which designates a syllogism in which
one of the premises is understood, and not expressed. Thus
we say in a mathematical proof, 'because the point C is the
center of the circle AB, AC is equal to BC;' not stating the
_major_,--that all lines drawn from the center of a circle
to the circumference are equal; or introducing it only by a
transient reference to the definition of a circle. But the
enthymeme is so constantly used in all habitual forms of
reasoning, that it does not occur to us as being anything
peculiar in mathematical works.
The propositions which are proved to be generally true are
termed _Theorems_: but when anything is required to be done,
as to draw a line or a circle under given conditions, this
proposition is a _Problem_. A theorem requires
demonstration; a problem, solution. And for both purposes
the mathematician usually makes a _Construction_. He directs
us to draw certain lines, circles, or other curves, on which
is to be founded his demonstration that his theorem is true,
or that his problem is solved. Sometimes, too, he
establishes some _Lemma_, or preparatory proposition, before
he proceeds to his main task; and often he deduces from his
demonstration some conclusion in addition to that which was
the professed object of his proposition; and this is termed
a _Corollary_.
These technical terms are noted here, not as being very
important, but in order that they may not sound {149}
strange and unintelligible if we should have occasion to use
some of them. There is, however, one technical distinction
more peculiar, and more important.
3. _Geometrical Analysis and Synthesis._--In geometrical
reasoning such as we have described, we introduce at every
step some new consideration; and it is by combining all
these considerations, that we arrive at the conclusion, that
is, the demonstration of the proposition. Each step tends to
the final result, by exhibiting some part of the figure
under a new relation. To what we have already proved, is
added something more; and hence this process is called
_Synthesis_, or _putting together_. The proof flows on,
receiving at every turn new contributions from different
quarters; like a river fed and augmented by many tributary
streams. And each of these tributaries flows from some
definition or axiom as its fountain, or is itself formed by
the union of smaller rivulets which have sources of this
kind. In descending along its course, the synthetical proof
gathers all these accessions into one common trunk, the
proposition finally proved.
But we may proceed in a different manner. We may begin from
the formed river, and ascend to its sources. We may take the
proposition of which we require a proof, and may examine
what the supposition of its truth implies. If this be true,
then something else may be seen to be true; and from this,
something else, and so on. We may often, in this way,
discover of what simpler propositions our theorem or
solution is compounded, and may resolve these in succession,
till we come to some proposition which is obvious. This is
geometrical _Analysis_. Having succeeded in this analytical
process, we may invert it; and may descend again from the
simple and known propositions, to the proof of a theorem, or
the solution of a problem, which was our starting-place.
This process resembles, as we have said, tracing a river to
its sources. As we ascend the stream, we perpetually meet
with bifurcations; and some sagacity is needed to enable us
to see which, in each case, is the main stream: but if we
proceed in our research, we {150} exhaust the unexplored
valleys, and finally obtain a clear knowledge of the place
whence the waters flow. _Analytical_ is sometimes confounded
with _symbolical_ reasoning, on which subject we shall make
a remark in the next chapter. The object of that chapter is
to notice certain other fundamental principles and ideas,
not included in those hitherto spoken of, which we find
thrown in our way as we proceed in our mathematical
speculations. It would detain us too long, and involve us in
subtle and technical disquisitions, to examine fully the
grounds of these principles; but the Mathematics hold so
important a place in relation to the inductive sciences,
that I shall briefly notice the leading ideas which the
ulterior progress of the subject involves.
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