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The directions of motion in the space of α due to rest in a point of β is called the 'β-direction in α' and the direction of motion in the space of β due to rest in a point of α is called the 'α-direction in β. Consider a motion in the space of α consisting of a certain velocity in the β-direction of α and a certain velocity at right-angles to it.
Streets proceed at right-angles with the river's course; and each street is lined with neat frame or brick houses, surrounding a square in such a manner that within each household has a sufficient garden.
§ 15. Returning then to Locke's comparison between the certainty of this argument and that which proves the sum of the angles of a triangle to be equal to two right-angles, I should say that there is a virtual, though not a formal, fallacy in his presentation. For mathematical science being confessedly but of relative significance, any comparison between the degree of certainty attained by reasoning upon so transcendental a subject as the present, and that of mathematical demonstrations regarding relative truth, must be misleading. In the present instance, the whole strain of the argument comes upon the adequacy of the proposed test of truth, viz., our being able to conceive it if true. Now, will any one undertake to say that this test of truth is of equivalent value when it is applied to a triangle and when it is applied to the Deity. In the one case we are dealing with a geometrical figure of an exceedingly simple type, with which our experience is well acquainted, and presenting a very limited number of relations for us to contemplate. In the other case we are endeavouring to deal with the summum genus of all mystery, with reference to which experience is quite impossible, and which in its mention contains all the relations that are to us unknown and unknowable. Here, then, is the oversight. Because men find conceivability a valid test of truth in the affairs of everyday life as it is easy to show
Neither the pillar, nor the arch, much less the vault, was a feature in their principal buildings, which affected straight lines, right-angles, and a massive construction, based upon the Egyptian.
This motion represents rest in the space of another time-system call it π. Rest in π will also be represented in the space of β by a certain velocity in the α-direction in β and a certain velocity at right-angles to this α-direction.
As the herd approaches us, it swings round its front at right-angles, and makes off westward. We dash forward, and divide it into two parties. We also separate, some of our hunters following one part of the herd, the others the remainder. The enthusiasm of our horses equals our own. Away we go; nothing stops us.
The lofty tower of its City Hall overlooked the whole panorama of the streets and avenues, which cut each other at right-angles, and in the midst of which appeared pleasant, verdant squares, while beyond appeared the Chinese quarter, seemingly imported from the Celestial Empire in a toy-box.
But far more terrible is it to behold, when fathoms down in the sea, you see some sulky whale, floating there suspended, with his prodigious jaw, some fifteen feet long, hanging straight down at right-angles with his body, for all the world like a ship's jib-boom.
It would be useless to stop where we were, so taking it for granted that the fugitives had gone eastward, we continued on at right-angles to their trail. We had not gone far when, crossing a piece of meadow, where from its exposure the snow had almost disappeared, Pat cried out, "There is something else worth having;" and, darting forward, he lifted up a long lasso composed of buffalo hide.
In the last connection it is interesting to note that the French Deperdussin firm produced a fabric wing-covering with extra strong threads run at right-angles through the fabric at intervals in order to limit the tearing to a defined area.
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