OF THE DEFINITIONS AND AXIOMS WHICH RELATE TO SPACE.
1. THE relations of space have been apprehended with
peculiar distinctness and clearness from the very first
unfolding of man's speculative powers. This was a
consequence of the circumstance which we have just noticed,
that the simplest of these relations, and those on which the
others depend, are seen by intuition. Hence, as soon as men
were led to speculate concerning the relations of space,
they assumed just principles, and obtained true results. It
is said that the science of _geometry_ had its origin in
Egypt, before the dawn of the Greek philosophy: but the
knowledge of the early Egyptians (exclusive of their
mythology) appears to have been purely practical; and,
probably, their geometry consisted only in some maxims of
_land-measuring_, which is what the term implies. The Greeks
of the time of Plato, had, however, not only possessed
themselves of many of the most remarkable elementary
theorems of the science; but had, in several instances,
reached the boundary of the science in its elementary form;
as when they proposed to themselves the problems of doubling
the cube and squaring the circle.
But the deduction of these theorems by a systematic process,
and the primary exhibition of the simplest principles
involved in the idea of space, which such a deduction
requires, did not take place, so far as we are aware, till a
period somewhat later. The _Elements of Geometry_ of Euclid,
in which this task was performed, are to this day the
standard work on the subject: the author of this work taught
mathematics with great applause at Alexandria, in the reign
of Ptolemy Lagus, {99} about 280 years before Christ. The
principles which Euclid makes the basis of his system have
been very little simplified since his time; and all the
essays and controversies which bear upon these principles,
have had a reference to the form in which they are stated by him.
2. _Definitions._--The first principles of Euclid's geometry
are, as the first principles of any system of geometry must
be, definitions and axioms respecting the various ideal
conceptions which he introduces; as straight lines, parallel
lines, angles, circles, and the like. But it is to be
observed that these definitions and axioms are very far from
being arbitrary hypotheses and assumptions. They have their
origin in the idea of space, and are merely modes of
exhibiting that idea in such a manner as to make it afford
grounds of deductive reasoning. The axioms are necessary
consequences of the conceptions respecting which they are
asserted; and the definitions are no less necessary
limitations of conceptions; not requisite in order to arrive
at this or that consequence; but necessary in order that it
may be possible to draw any consequences, and to establish
any general truths.
For example, if we rest the end of one straight staff upon
the middle of another straight staff, and move the first
staff into various positions, we, by so doing, alter the
angles which the first staff makes with the other to the
right hand and to the left. But if we place the staff in
that special position in which these two angles are equal,
each of them is a right angle, according to Euclid; and this
is the _definition_ of a right angle, except that Euclid
employs the abstract conception of straight lines, instead
of speaking, as we have done, of staves. But this selection
of the case in which the two angles are equal is not a mere
act of caprice; as it might have been if he had selected a
case in which these angles are unequal in any proportion.
For the consequences which can be drawn concerning the cases
of unequal angles, do not lead to general truths, without
some reference to that peculiar case in which the angles are
equal: and thus it becomes necessary to {100} single out and
define that special case, marking it by a special phrase.
And this definition not only gives complete and distinct
knowledge what a right angle is, to any one who can form the
conception of an angle in general; but also supplies a
principle from which all the properties of right angles may
be deduced.
3. _Axioms._--With regard to other conceptions also, as
circles, squares, and the like, it is possible to lay down
definitions which are a sufficient basis for our reasoning,
so far as such figures are concerned. But, besides these
definitions, it has been found necessary to introduce
certain axioms among the fundamental principles of geometry.
These are of the simplest character; for instance, that two
straight lines cannot cut each other in more than one point,
and an axiom concerning parallel lines. Like the
definitions, these axioms flow from the Idea of Space, and
present that idea under various aspects. They are different
from the definitions; nor can the definitions be made to
take the place of the axioms in the reasoning by which
elementary geometrical properties are established. For
example, the definition of parallel straight lines is, that
they are such as, however far continued, can never meet:
but, in order to reason concerning such lines, we must
further adopt some axiom respecting them: for example, we
may very conveniently take this axiom; that two straight
lines which cut one another are not both of them parallel to
a third straight line[1\2]. The definition and the axiom are
seen to be inseparably connected by our intuition of the
properties of space; but the axiom cannot be proved from the
definition, by any rigorous deductive demonstration. And if
we were to take any other definition of two parallel
straight lines, (as that they are both perpendicular to a
third straight line,) we should still, at some point or
other of our progress, fall in with the same difficulty of
demonstratively establishing their properties without some
further assumption.
[Note 1\2: This axiom is simpler and more convenient than
that of Euclid. It is employed by the late Professor
Playfair in his _Geometry_.]
{101} 4. Thus the elementary properties of figures, which
are the basis of our geometry, are necessary results of our
Idea of Space; and are connected with each other by the
nature of that idea, and not merely by our hypotheses and
constructions. Definitions and axioms must be combined, in
order to express this idea so far as the purposes of
demonstrative reasoning require. These verbal enunciations
of the results of the idea cannot be made to depend on each
other by logical consequence; but have a mutual dependence
of a more intimate kind, which words cannot fully convey. It
is not possible to resolve these truths into certain
_hypotheses_, of which all the rest shall be the necessary
logical consequence. The necessity is not hypothetical, but
intuitive. The axioms require not to be granted, but to be
seen. If any one were to assent to them without seeing them
to be true, his assent would be of no avail for purposes of
reasoning: for he would be also unable to see in what cases
they might be applied. The clear possession of the Idea of
Space is the first requisite for all geometrical reasoning;
and this clearness of idea may be tested by examining
whether the axioms offer themselves to the mind as evident.
5. The necessity of ideas added to sensations, in order to
produce knowledge, has often been overlooked or denied in
modern times. The ground of necessary truth which ideas
supply being thus lost, it was conceived that there still
remained a ground of necessity in definitions;--that we
might have necessary truths, by asserting especially what
the definition implicitly involved in general. It was held,
also, that this was the case in geometry:--that all the
properties of a circle, for instance, were implicitly
contained in the definition of a circle. That this alone is
not the ground of the necessity of the truths which regard
the circle,--that we could not in this way unfold a
definition into proportions, without possessing an intuition
of the relations to which the definition led,--has already
been shown. But the insufficiency of the above account of
the grounds of necessary geometrical truth appeared in
another way also. It was found impossible to lay {102} down
a system of definitions out of which alone the whole of
geometrical truth could be evolved. It was found that axioms
could not be superseded. No definition of a straight line
could be given which rendered the axiom concerning straight
lines superfluous. And thus it appeared that the source of
geometrical truths was not definition alone; and we find in
this result a confirmation of the doctrine which we are here
urging, that this source of truth is to be found in the form
or conditions of our perception;--in the idea which we
unavoidably combine with the impressions of sense;--in the
activity, and not in the passivity of the mind[2\2].
[Note 2\2: I formerly stated views similar to these in some
'Remarks' appended to a work which I termed _The Mechanical
Euclid_, published in 1837. These Remarks, so far as they
bear upon the question here discussed, were noticed and
controverted in No. 135 of the _Edinburgh Review_. As an
examination of the reviewer's objections may serve further
to illustrate the subject, I shall annex to this chapter an
answer to the article to which I have referred.]
6. This will appear further when we come to consider the
mode in which we exercise our observation upon the relations
of space. But we may, in the first place, make a remark
which tends to show the connexion between our conception of
a straight line, and the axiom which is made the foundation
of our reasonings concerning space. The axiom is this;--that
two straight lines, which have both their ends joined,
cannot have the intervening parts separated so as to inclose
a space. The necessity of this axiom is of exactly the same
kind as the necessity of the definition of a right angle, of
which we have already spoken. For as the line standing on
another makes _right angles_ when it makes the angles on the
two sides of it equal; so a line is a _straight line_ when
it makes the two portions of space, on the two sides of it,
similar. And as there is only a single position of the line
first mentioned, which can make the angles equal, so there
is only a single form of a line which can make the spaces
near the line similar on one side and on the other: and
{103} therefore there cannot be two straight lines, such as
the axiom describes, which, between the same limits, give
two different boundaries to space thus separated. And thus
we see a reason for the axiom. Perhaps this view may be
further elucidated if we take a leaf of paper, double it,
and crease the folded edge. We shall thus obtain a straight
line at the folded edge; and this line divides the surface
of the paper, as it was originally spread out, into two
similar spaces. And that these spaces are similar so far as
the fold which separates them is concerned, appears from
this;--that these two parts coincide when the paper is
doubled. And thus a fold in a sheet of paper at the same
time illustrates the definition of a straight line according
to the above view, and confirms the axiom that two such
lines cannot inclose a space.
If the separation of the two parts of space were made by any
other than a straight line; if, for instance, the paper were
cut by a concave line; then, on turning one of the parts
over, it is easy to see that the edge of one part being
concave one way, and the edge of the other part concave the
other way, these two lines would enclose a space. And each
of them would divide the whole space into two portions which
were not similar; for one portion would have a concave edge,
and the other a convex edge. Between any two points, there
might be innumerable lines drawn, some, convex one way, and
some, convex the other way; but the straight line is the
line which is not convex either one way or the other; it is
the single medium standard from which the others may deviate
in opposite directions.
Such considerations as these show sufficiently that the
singleness of the straight line which connects any two
points is a result of our fundamental conceptions of space.
But yet the above conceptions of the similar form of the two
parts of space on the two sides of a line, and of the form
of a line which is intermediate among all other forms, are
of so vague a nature, that they cannot fitly be made the
basis of our elementary geometry; and they are far more
conveniently replaced, as they have been in almost all
treatises of {104} geometry, by the axiom, that two straight
lines cannot inclose a space.
7. But we may remark that, in what precedes, we have
considered space only under one of its aspects:--as a plane.
The sheet of paper which we assumed in order to illustrate
the nature of a straight line, was supposed to be perfectly
_plane_ or _flat_: for otherwise, by folding it, we might
obtain a line not straight. Now this assumption of a plane
appears to take for granted that very conception of a
straight line which the sheet was employed to illustrate;
for the definition of a plane given in the Elements of
Geometry is, that it is a surface on which lie all straight
lines drawn from one point of the surface to another. And
thus the explanation above given of the nature of a straight
line,--that it divides a plane space into similar portions
on each side,--appears to be imperfect or nugatory.
To this we reply, that the explanation must be rendered
complete and valid by deriving the conception of a plane
from considerations of the same kind as those which we
employed for a straight line. Any portion of solid space may
be divided into two portions by surfaces passing through any
given line or boundaries. And these surfaces may be convex
either on one side or on the other, and they admit of
innumerable changes from being convex on one side to being
convex on the other in any degree. So long as the surface is
convex either way, the two portions of space which it
separates are not similar, one having a convex and the other
a concave boundary. But there is a certain intermediate
position of the surface, in which position the two portions
of space which it divides have their boundaries exactly
similar. In this position, the surface is neither convex nor
concave, but plane. And thus a plane surface is determined
by this condition--of its being that single surface which is
the intermediate form among all convex and concave surfaces
by which solid space can be divided,--and of its separating
such space into two portions, of which the boundaries,
though they are the same surface in two opposite positions,
are exactly similar. {105}
Thus a plane is the simplest and most symmetrical boundary
by which a solid can be divided; and a straight line is the
simplest and most symmetrical boundary by which a plane can
be separated. These conceptions are obtained by considering
the boundaries of an interminable space, capable of
imaginary division in every direction. And as a limited
space may be separated into two parts by a plane, and a
plane again separated into two parts by a straight line, so
a line is divided into two portions by a point, which is the
common boundary of the two portions; the end of the one and
the beginning of the other portion having itself no
magnitude, form, or parts.
8. The geometrical properties of planes and solids are
deducible from the first principles of the Elements, without
any new axioms; the definition of a plane above
quoted,--that all straight lines joining its points lie in
the plane,--being a sufficient basis for all reasoning upon
these subjects. And thus, the views which we have presented
of the nature of space being verbally expressed by means of
certain definitions and axioms, become the groundwork of a
long series of deductive reasoning, by which is established
a very large and curious collection of truths, namely, the
whole science of Elementary Plane and Solid Geometry.
This science is one of indispensable use and constant
reference, for every student of the laws of nature; for the
relations of space and number are the _alphabet_ in which
those laws are written. But besides the interest and
importance of this kind which geometry possesses, it has a
great and peculiar value for all who wish to understand the
foundations of human knowledge, and the methods by which it
is acquired. For the student of geometry acquires, with a
degree of insight and clearness which the unmathematical
reader can but feebly imagine, a conviction that there are
necessary truths, many of them of a very complex and
striking character; and that a few of the most simple and
self-evident truths which it is possible for the mind of man
to apprehend, may, by systematic deduction, lead to the most
remote and unexpected results. {106}
In pursuing such philosophical researches as that in which
we are now engaged, it is of great advantage to the
speculator to have cultivated to some extent the study of
geometry; since by this study he may become fully aware of
such features in human knowledge as those which we have
mentioned. By the aid of the lesson thus learned from the
contemplation of geometrical truths, we have been
endeavouriug to establish those further doctrines;--that
these truths are but different aspects of the same
Fundamental Idea, and that the grounds of the necessity
which these truths possess reside in the Idea from which
they flow, this Idea not being a derivative result of
experience, but its primary rule. When the reader has
obtained a clear and satisfactory view of these doctrines,
so far as they are applicable to our knowledge concerning
space, he has, we may trust, overcome the main difficulty
which will occur in following the course of the speculations
now presented to him. He is then prepared to go forwards
with us; to see over how wide a field the same doctrines are
applicable: and how rich and various a harvest of knowledge
springs from these seemingly scanty principles.
But before we quit the subject now under our consideration,
we shall endeavour to answer some objections which have been
made to the views here presented; and shall attempt to
illustrate further the active powers which we have ascribed
to the mind.
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