.
It will be found that, from the seven properties which we have assigned to the relation "between," it can be deduced that the relation "to the left of," as above defined, is a serial relation as we defined that term. It is important to notice that nothing in the definitions or the argument depends upon our meaning by "between" the actual relation of that name which occurs in empirical space: any three-term relation having the above seven purely formal properties will serve the purpose of the argument equally well.
Cyclic order, such as that of the points on a circle, cannot be generated by means of three-term relations of "between." We need a relation of four terms, which may be called "separation of couples." The point may be illustrated by considering a journey round the world. One may go from England to New Zealand by way of Suez or by way of San Francisco; we cannot say definitely that either of these two places is "between" England and New Zealand. But if a man chooses that route to go round the world, whichever way round he goes, his times in England and New Zealand are separated from each other by his times in Suez and San Francisco, and conversely. Generalising, if we take any four points on a circle, we can separate them into two couples, say
and
and
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