OF THE FOUNDATIONS OF THE HIGHER MATHEMATICS.
1. _The Idea of a Limit._--THE general truths concerning
relations of space which depend upon the axioms and
definitions contained in Euclid's _Elements_, and which
involve only properties of straight lines and circles, are
termed Elementary Geometry: all beyond this belongs to the
Higher Geometry. To this latter province appertain, for
example, all propositions respecting the lengths of any
portions of curve lines; for these cannot be obtained by
means of the principles of the Elements alone. Here then we
must ask to what other principles the geometer has recourse,
and from what source these are drawn. Is there any origin of
geometrical truth which we have not yet explored?
The _Idea of a Limit_ supplies a new mode of establishing
mathematical truths. Thus with regard to the length of any
portion of a curve, a problem which we have just mentioned;
a curve is not made up of straight lines, and therefore we
cannot by means of any of the doctrines of elementary
geometry measure the length of any curve. But we may make up
a figure nearly resembling any curve by putting together
many short straight lines, just as a polygonal building of
very many sides may nearly resemble a circular room. And in
order to approach nearer and nearer to the curve, we may
make the sides more and more small, more and more numerous.
We may then possibly find some mode of measurement, some
relation of these small lines to other lines, which is not
disturbed by the multiplication of the sides, however far it
be carried. And thus, we may do what is equivalent to
measuring the curve itself; for by multiplying the {152}
sides we may approach more and more closely to the curve
till no appreciable difference remains. The curve line is
the _Limit_ of the polygon; and in this process we proceed
on the _Axiom_, that 'What is true up to the Limit is true
at the Limit.'
This mode of conceiving mathematical magnitudes is of wide
extent and use; for every curve may be considered as the
limit of some polygon; every varied magnitude, as the limit
of some aggregate of simpler forms; and thus the relations
of the elementary figures enable us to advance to the
properties of the most complex cases.
A Limit is a peculiar and fundamental conception, the use of
which in proving the propositions of the Higher Geometry
cannot be superseded by any combination of other hypotheses
and definitions[14\2]. The axiom just noticed, that what is
true up to the limit is true at the limit, is involved in
the very conception of a Limit: and this principle, with its
consequences, leads to all the results which form the
subject of the higher mathematics, whether proved by the
consideration of evanescent triangles, by the processes of
the Differential Calculus, or in any other way.
[Note 14\2: This assertion cannot be fully proved and
illustrated without a reference to mathematical reasonings
which would not be generally intelligible. I have shown the
truth of the assertion in my _Thoughts on the Study of
Mathematics_, annexed to the _Principles of English
University Education_. The proof is of this kind:--The
ultimate equality of an arc of a curve and the corresponding
periphery of a polygon, when the sides of the polygon are
indefinitely increased in number, is _evident_. But this
truth cannot be proved from any other axiom. For if we take
the supposed axiom, that a curve is always less than the
including broken line, this is not true, except with a
condition; and in tracing the import of this condition, we
find its necessity becomes evident only when we introduce a
reference to a Limit. And the same is the case if we attempt
to supersede the notion of a Limit in proving any other
simple and evident proposition in which that notion is
involved. Therefore these evident truths are _self_-evident,
_in virtue of the Idea of a Limit_.]
The ancients did not expressly introduce this conception of
a Limit into their mathematical reasonings; although in the
application of what is termed the {153} _Method of
Exhaustions_, (in which they show how to _exhaust_ the
_difference_ between a polygon and a curve, or the like,)
they were in fact proceeding upon an obscure apprehension of
principles equivalent to those of the Method of Limits. Yet
the necessary fundamental principle not having, in their
time, been clearly developed, their reasonings were both
needlessly intricate and imperfectly satisfactory. Moreover
they were led to put in the place of axioms, assumptions
which were by no means self-evident; as when Archimedes
assumed, for the basis of his measure of the circumference
of the circle, the proposition that a circular arc is
necessarily less than two lines which inclose it, joining
its extremities. The reasonings of the older mathematicians,
which professed to proceed upon such assumptions, led to
true results in reality, only because they were guided by a
latent reference to the limiting case of such assumptions.
And this latent employment of the conception of a Limit,
reappeared in various forms during the early period of
modern mathematics; as for example, in the _Method of
Indivisibles_ of Cavalleri, and the _Characteristic
Triangle_ of Barrow; till at last, Newton distinctly
referred such reasonings to the conception of a Limit, and
established the fundamental principles and processes which
that conception introduces, with a distinctness and
exactness which required little improvement to make it as
unimpeachable as the demonstrations of geometry. And when
such processes as Newton thus deduced from the conception of
a Limit, are represented by means of general algebraical
symbols instead of geometrical diagrams, we have then before
us the _Method of Fluxions_, or the _Differential Calculus_;
a mode of treating mathematical problems justly considered
as the principal weapon by which the splendid triumphs of
modern mathematics have been achieved.
2. _The Use of General Symbols._--The employment of
algebraical symbols, of which we have just spoken, has been
another of the main instruments to which the successes of
modern mathematics are owing. And here again the processes
by which we obtain our {154} results depend for their
evidence upon a fundamental conception,--the conception of
_arbitrary symbols_ as the _Signs_ of quantity and its
relations; and upon a corresponding axiom, that 'The
interpretation of such symbols must be perfectly general.'
In this case, as in the last, it was only by degrees that
mathematicians were led to a just apprehension of the
grounds of their reasoning. For symbols were at first used
only to represent numbers considered with regard to their
numerical properties; and thus the science of Algebra was
formed. But it was found, even in cases belonging to common
algebra, that the symbols often admitted of an
interpretation which went beyond the limits of the problem,
and which yet was not unmeaning, since it pointed out a
question closely analogous to the question proposed. This
was the case, for example, when the answer was a _negative
quantity_; for when Descartes had introduced the mode of
representing curves by means of algebraical relations among
the symbols of the _co-ordinates_, or distances of each of
their points from fixed lines, it was found that negative
quantities must be dealt with as not less truly significant
than positive ones. And as the researches of mathematicians
proceeded, other cases also were found, in which the
symbols, although destitute of meaning according to the
original conventions of their institution, still pointed out
truths which could be verified in other ways; as in the
cases in which what are called _impossible quantities_
occur. Such processes may usually be confirmed upon other
principles, and the truth in question may be established by
means of a demonstration in which no such seeming fallacies
defeat the reasoning. But it has also been shown in many
such cases, that the process in which some of the steps
appear to be without real meaning, does in fact involve a
valid proof of the proposition. And what we have here to
remark is, that this is not true accidentally or partially
only, but that the results of systematic symbolical
reasoning must _always_ express general truths, by their
nature; and do not, for their justification, require each of
the steps of the process to represent {155} some definite
operation upon quantity. _The absolute universality of the
interpretation of symbols_ is the fundamental principle of
their use. This has been shown very ably by Dr. Peacock in
his _Algebra_. He has there illustrated, in a variety of
ways, this principle: that 'If general symbols express an
identity when they are supposed to be of any special nature,
they must also express an identity when they are general in
their nature.' And thus, this universality of symbols is a
principle in addition to those we have already noticed; and
is a principle of the greatest importance in the formation
of mathematical science, according to the wide generality
which such science has in modern times assumed.
3. _Connexion of Symbols and Analysis._--Since in our
symbolical reasoning our symbols thus reason for us, we do
not necessarily here, as in geometrical reasoning, go on
adding carefully one known truth to another, till we reach
the desired result. On the contrary, if we have a theorem to
prove or a problem to solve which can be brought under the
domain of our symbols, we may at once state the given but
unproved truth, or the given combination of unknown
quantities, in its symbolical form. After this first
process, we may then proceed to trace, by means of our
symbols, what other truth is involved in the one just
stated, or what the unknown symbols must signify; resolving
step by step the symbolical assertion with which we began,
into others more fitted for our purpose. The former process
is a kind of _synthesis_, the latter is termed _analysis_.
And although symbolical reasoning does not necessarily imply
such analysis; yet the connexion is so familiar, that the
term _analysis_ is frequently used to designate symbolical
reasoning.
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