Mr Norbert Wiener, A Contribution to the Theory, etc. 441 
A Contribution to the Theory of Relative Position*. By NORBERT 
WieENER, Ph.D. (Communicated by Mr G. H. Hardy.) 
[Recewed 14 March 1914. } 
The theory of relations is one of the most interesting depart- 
ments of the new mathematical logic. The relations which have 
been most thoroughly studied are the series: that 1s, relations 
which are contained in diversity, transitive, and connected or, in 
Mr Russell’s symbolism, those relations R of which the following 
proposition is true: 
RGJ.RER. Rv Rol POR=ORF OR. 
Cantor, Dedekind, Frege, Schroder, Burali-Forti, Huntington, 
Whitehead, and Russell, are among those who have helped to 
give us an almost exhaustive account of the more fundamental 
properties of series. There is a class of relations closely allied to 
series, however, which has received very scant attention from the 
mathematical logicians. Examples of the sort of relation to which 
I am referring are the relation between two events in time when 
one completely precedes the other, or the relation between two 
intervals on a line when one lies to the left of the other, and does 
not overlap it, or, in general, the relation between two stretches 
a and #, of terms of a series A, when any term lying in a bears 
the relation R to any term lying in @. Relations of this sort, 
which I shall call relations of complete sequence, differ in general 
from series in not being connected: that is, for example, it is not 
necessary that of two distinct events, each of which wholly 
precedes or follows some other event, one should wholly precede 
the other, for the times of their occurrence may overlap. But in 
all the instances we have given, the relation of complete sequence 
is closely bound up with some serial relation: the relation of 
succession between the events of time is intimately related to the 
series of its instants, the relation between two intervals on a line 
one of which lies completely to the other’s left is intimately related 
to the series of the points on the line, and so on. These con- 
siderations lead us to the general questions, (1) what are the 
formal properties which characterise relations of the sort we have 
* The subject of this paper was suggested to me by Mr Bertrand Russell, and 
the paper itself is the result of an attempt to simplify and generalize certain notions 
used by him in his treatment of the relation between the series of events and the 
series of instants. 
29—5 
