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An I pose dat we rect a tent on de sho oh dis yah island, and hab de banket come off in fust chop style." "The island!" cried Tom, in horror. "What! the island? Breakfast on the island? What a horrible proposal! Look here, captain. Can't we get away from this?" "Get away from this?" repeated the captain, in mild surprise. "Yes," said Tom.
An abstractive element which belongs to a punct has the simplest type of position in M, an abstractive element which belongs to a rect but not to a punct has a more complex quality of position, an abstractive element which belongs to a level and not to a rect has a still more complex quality of position, and finally the most complex quality of position belongs to an abstractive element which belongs to a volume and not to a level.
Any rect in M which intersects all these levels in its set of puncts, thereby receives for its puncts an order of position on it. So spatial order is derivative from temporal order. Furthermore there are alternative time-systems, but there is only one definite spatial order in each instantaneous space.
Let the satisfaction of the condition σ by an abstractive set mean that the two given event-particles and the event-particles lying between them on the rect all lie in every event belonging to the abstractive set. The group of σ-primes, where σ has this meaning, form an abstractive element. Such abstractive elements are rectilinear routes.
Then the straight line r of space α is in the direction of β in space α, and the rect ρ is the representative in the moment M of the line r of space α. The direction of ρ in the instantaneous space of the moment M is the direction of β in M, where M is a moment of time-system α.
Thus in the four-dimensional geometry of all event-particles there is a two-dimensional locus which is the locus of all event-particles on points lying on the straight line r. I will call this locus of event-particles the matrix of the straight line r. A matrix intersects any moment in a rect. Thus the matrix of r intersects the moment M in a rect ρ.
Rectilinear routes are momental routes and stations are vagrant routes. Rectilinear routes are routes which in a sense lie in rects. Any two event-particles on a rect define the set of event-particles which lie between them on that rect.
For example, we see a stretch of road and a lorry moving along it. The instantaneously seen road is a portion of the rect ρ of course only an approximation to it. The lorry is the moving object. But the road as seen is never traversed.
M. S. Optimi parentis EDWARDI YOUNG, LL.D. Hujus Ecclesiae rect. et Elizabethae faem. praenob Conjugis ejus amantissimae Pio et gratissimo animo hoc marmor posuit F. Y. Filius superstes. Is it not strange that the author of the "Night Thoughts" has inscribed no monument to the memory of his lamented wife? Yet what marble will endure as long as the poems?
We shall find in the next lecture that it is from this symmetry that the theory of congruence is deduced. The theory of perpendicularity in the timeless space of any time-system α follows immediately from this theory of perpendicularity in each of its instantaneous spaces. Let ρ be any rect in the moment M of α and let λ be a level in M which is perpendicular to ρ.
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