The weak point of the Euclidean definition of a straight line is, that nothing has been deduced from it. The notion expressed by the phrases ‘evenly,’ or ‘evenly placed,’ requires definition. The definition should be such that the uniqueness of the straight segment between two points can be deduced from it. Neither of these demands has ever been satisfied, with the result that in modern times the notion of ‘straightness’ has been based on that of measurement. A straight line has, in modern times, been defined as the shortest distance between two points. In the classic geometry, the converse procedure was adopted, and measurement presupposed straight lines. But, with the modern definition, the notion of the ‘shortest distance’ in its turn requires explanation.’ This notion is practically defined to mean the line which is the route of certain physical occurrences.
In this section it will be shown that the gap in the old
1Cf. Part IV, Ch. V, on ‘Measurement.’
462 Processuan Dsr Eady
classical theory can be remedied. Straight lines will be defined in terms of the extensive notions, developed in the preceding chapter; and the uniqueness of the straight line joining two points will be proved to follow from the terms of the definition.
A class of ‘oval’ regions must first be defined. Now the only weapon which we have for this definition is the notion of regions which overlap with a unique intersect (cf. Def. 6 of previous chapter). It is evidently a prop- erty of a pair of ovals that they can only overlap with a unique intersect. But it is equally evident that some regions which are not ovals also overlap with a unique intersect. However the class of ovals has the property that any region, not a member of it, intersects some ovals with multiple intersects. Also sub-sets of ovals can be found satisfying various conditions.
Thus we proceed to define a class whose regions shall have those relations to each other, and to other regions, which we ascribe to the class of ovals. In other words we cannot define a single oval, but we can define a class of ovals. Such a class will be called ‘ovate.’ The defini- tion of an ovate class proceeds by enumerating all those peculiar properties possessed by individual members of the class, or by sub-sets of members of the class. It will be found in the course of this enumeration that an exten- sive continuum which possesses an ovate class is dimen- sional in respect to that class. Thus existence of straight lines in an extensive continuum is bound up with the dimensional character of the continuum; and both char- acteristics are relative to a particular ovate class of regions in the continuum. It seems probable that an extensive continuum will possess only one ovate class. But I have not succeeded in proving that property; nor is it necessary for the argument.
A preliminary definition is convenient:
Definition 0.1. An ‘ovate abstractive set’ is an abstrac-
tive set whose members all belong to the complete ovate
class under consideration.
POL ATL Over 463
The characteristics of an ovate class will be divided
into two groups: (a) the group of non-abstractive condi-
tions, and (b) the group of abstractive conditions.
Definition 1. A class of regions is called ‘ovate,’ when
it satisfies the conditions belonging to the two following
groups, (a) and (b):
(a) The Non-Abstractive Group
(1) Any two overlapping regions of the ovate class
have a unique intersect which also belongs to that ovate
class.
(ii) Any region, not a member of the ovate class, over-
laps some members of that class with ‘multiple intersec-
tion’ (cf. Def. 6 of previous chapter).
(iii) Any member of the ovate class overlaps some regions, not of that class, with multiple intersection.
(iv) Any pair of members of the ovate class, which are externally connected, have their surfaces touching either in a ‘complete locus’ of points (cf. Ch. III, Def. 23 and Ass. 33), or in a single point.
(v) Any region, not belonging to the ovate class, is
externally connected with some member of that class so
that their surfaces touch in a set of points which does
not form a ‘complete locus.’
(vi) Any member of the ovate class is externally con-
nected with some region not of that class so that their
surfaces touch in a set of points which does not form a
‘complete locus.’
(vii) Any finite number of regions are jointly included in some member of the ovate class.
(viii) If A and B be members of the ovate class, and
A include B, then there are members of the class which
include B and are included in A.
(ix) There are dissections (cf. Def. 4 of the previous
chapter) of every member of the ovate class, which con-
sist wholly of members of that class; and there are dis-
sections consisting wholly or partly of members not
belonging to that class.
464 Process and Reality
(b) The Abstractive Group
(i) Among the members of any point, there are ovate abstractive sets.
(ii) If any set of two, or of three, or of four, points be considered, there are abstractive sets ‘prime’ in reference to the twofold condition, (a) of covering the points in question, and (b) of being equivalent to an ovate abstrac- tive set.
(iii) These are sets of five points such that no abstrac- tive set exists prime in reference to the twofold condition, (a) of covering the points in question, and (b) of being equivalent to an ovate abstractive set.
By reason of the definitions of this latter group, the
extensive continuum in question is called ‘four-dimen-
sional.’ Analogously, an extensive continuum of any
number of dimensions can be defined. The physical
extensive continuum with which we are concerned in this
cosmic epoch is four-dimensional. Notice that the prop-
erty of being ‘dimensional’ is relative to a particular
ovate class in the extensive continuum. There may be
‘ovate’ classes satisfying all the conditions with the ex-
ception of the ‘dimensional’ conditions. Also a contin-
uum may have one number of dimensions relating to one
ovate class, and another number of dimensions relatively
to another ovate class.
Possibly physical laws, of the type presupposing con-
tinuity, depend on the interwoven properties of two, or
more, distinct ovate classes.
Assumption 1. In the extensive continuum of the
present epoch there is at least one ovate class, with the
characteristics of the two groups, (a) and (b), of the
previous section.
Defintion 2. One such ovate class will be denoted by
a: all definitions will be made relatively to this selected
ovate class.
Peat Loc! 465
It is indifferent to the argument whether or no there be an alternative ovate class. If there be, the derivative entities defined in reference to this alternative class are entirely different to those defined in reference to q. It is sufficient for us, that one such class interests us by the importance of its physical relations.
Assumption 2. If two abstractive sets are prime in ref-
erence to the same twofold condition, (a) of covering a
given group of points, and (b) of being equivalent to
some ovate abstractive set, then they are equivalent.
By reason of the importance of this proposition a proof is given.
Proof. The two abstractive sets are either equivalent to the same ovate abstractive set, or to different ovate abstractive sets. In the former alternative, the required conclusion is obvious. In the latter alternative, let and v be the two different ovate abstractive sets. Each of these sets, p and y, satisfies the twofold condition. We have to prove that they are equivalent to each other. Let M and N be any regions belonging to p and y respectively. Then since the convergent portions of the abstractive sets belonging to the various points of the given group must ultimately consist of regions all lying in M and all lying in N, it follows that M and N intersect. But, being oval, M and N have only one intersect, and all the points in ques- tion must be situated in it. Also this intersect is oval. Hence, by selecting such intersects, a third abstractive set can be found which satisfies the twofold condition and is covered both by y and by y. But since y and vy are prime in reference to this condition, they are both of them equivalent to this third abstractive class. Hence they are equivalent to each other. Q.E.D.
Corollary. It follows that all abstractive sets,
prime with respect to the same twofold condition of this
type, belong to one geometrical element.
Definition 3. The single geometrical element defined,
as in the enunciation of Assumption 2, by a set of two
points is called a ‘straight’ segment between those end-
466 Process and Reality
points. If the set comprise more than two points, the geometrical element is called ‘flat.’ ‘Straight’ segments are also included under the designation ‘flat geometrical elements.’
If a set of points define a flat geometrical element, as
in the enunciation of Assumption 2, it may happen that
the same geometrical element is defined by some sub-set
of those points. Hence we have the following defi-
nition:
Definition 4. A set of points, defining a flat geometri-
cal element, is said to be in its lowest terms when it con-
tains no sub-set defining the same flat geometrical
element.
Assumption 3. No two sets of a finite number of
points, both in their lowest terms, define the same flat
geometrical element.
Definition & The locus of points incident in a ‘straight
segment’ is called the ‘straight line’ between the end-
points of the segment.
Definition 6. The locus of points incident in a flat
geometrical element is called the ‘content’ of that ele-
ment. It is also called a ‘flat locus.’
Assumption 4. If any sub-set of points lie in a flat
locus, that sub-set also defines a flat locus contained
within the given locus.
Definition 6. A complete straight line is a locus of points such that, (i) the straight line joining any two members of the locus lies wholly within the locus, (ii) every sub-set in the locus, which is in its lowest terms, consists of a pair of points, (ili) no points can be added to the locus without loss of one, or both, of the charac- teristics (i) and (ii).
Definition 7. A triangle is the flat locus defined by
three points which are not collinear. The three points
are the angular points of the triangle.
Definition 8. A plane is a locus of non-collinear points such that, (1) the triangle defined by any three non-col- linear members of the locus lies wholly within the locus,
PUATMMO Ci 467
(ii) any finite number of points in the locus lie in some
triangle wholly contained in the locus, (ili) no set of
points can be added to the locus without loss of one, or
both, of the characteristics (i) and (ii).
Definition 9. A tetrahedron is the flat locus defined by
four points which are not coplanar. The four points are
called the corners of the tetrahedron.
Definition 10. A three-dimensional flat space is a locus of non-coplanar points such that, (i) the tetrahedron defined by any four non-coplanar points of the locus lies wholly within the locus, (ii) any finite number of points in the locus lie in some tetrahedron wholly contained in the locus, (i11) no set of points can be added to the locus without the loss of one, or both, of the characteristics (i) and (ii).
Any further development of definitions and propo-
sitions will lead to mathematical details irrelevant
to our immediate purposes. It suffices to have
proved that characteristic properties of straight lines,
planes, and three-dimensional flat spaces are discover-
able in the extensive continuum without any recourse
to measurement. The systematic character of a con-
tinuum depends on its possession of one or more ovate
classes. Here, the particular case of a ‘dimensional’
ovate class has been considered.
The importance of the notion of ‘external connection’ requires further discussion.
First, there is a purely geometrical question to be noted. The theory of the external connection of oval regions throws light on the Euclidean concept of ‘even- ness.’ A pair of ovals (cf. Sect. III) can only be exter- nally connected in a ‘complete locus,’ or in a single point. We now consider that species of ‘complete loci’ which can be the points common to the surfaces of a pair of ovals externally connected. We exclude the case of one- point contact. The species seems to have what the
468 Process and Reality
Greeks meant by their term ‘even’ ({ooc). On either side of such a locus, there is the interior of one oval and the exterior of another oval, so that the locus is ‘even’ in respect to the contrasted notions of ‘concavity’ and ‘convexity.’ It is an extra ‘assumption’—provable or otherwise according to the particular logical develop- ment of the subject which may have been adopted— that all ‘even’ loci are ‘flat,’ and that all ‘flat’ loci are ‘even.’