The mathematician does not need to go through all the possible tests in detail; he knows that the complete compensation is inherent in the fundamental laws of nature, and so must occur in every case. So if any suggestion is made of a device for detecting these effects, he starts at once to look for the fallacy which must surely be there. Our motion through the aether may be very much less than the value here adopted, and the changes of length may be very small; but the essential point is that they escape notice, not because they are small (if they are small), but because from their very nature they are undetectable.
There is a remarkable reciprocity about the effects of motion on length, which can best be illustrated by another example. Suppose that by development in the powers of aviation, a man flies past us at the rate of 161,000 miles a second. We shall suppose that he is in a comfortable travelling conveyance in which he can move about, and act normally and that his length is in the direction of the flight. If we could catch an instantaneous glimpse as he passed, we should see a figure about three feet high, but with the breadth and girth of a normal human being. And the strange thing is that he would be sublimely unconscious of his own undignified appearance. If he looks in a mirror in his conveyance, he sees his usual proportions; this is because of the contraction of his retina, or the distortion by the moving mirror, as already explained. But when he looks down on us, he sees a strange race of men who have apparently gone through some flattening-out process; one man looks barely 10 inches across the shoulders, another standing at right angles is almost "length and breadth, without thickness." As they turn about they change appearance like the figures seen in the old-fashioned convex-mirrors. If the reader has watched a cricket-match through a pair of prismatic binoculars, he will have seen this effect exactly.
It is the reciprocity of these appearances that each party should think the other has contracted that is so difficult to realise. Here is a paradox beyond even the imagination of Dean Swift. Gulliver regarded the Lilliputians as a race of dwarfs; and the Lilliputians regarded Gulliver as a giant. That is natural, If the Lilliputians had appeared dwarfs to Gulliver,
