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 geometry. But what is not so familiar to mathematicians is that the restoration of the travelling particle to the point from which it originally started need not involve a journey of infinite length. If we assume Euclid's twelfth axiom to be true, then no doubt the traveller can only get back to the point from which he started as the result of a journey of infinite length which will have occupied an infinite time. But now suppose that Euclid's twelfth axiom be not true, or suppose that, what comes to the same thing, the three angles of a triangle are not indeed equal to two right angles, then the journey may require neither an infinite lapse of time nor an infinitely great speed before the traveller regains his original position, even though he be moving in a straight line all the time. Space is thus clearly finite; for a particle travelling in a straight line with uniform speed in the same direction is never able to get beyond a certain limited distance from the original position, to which it will every now and then return.

Those who remember their Euclid may be horror-struck at the heresy which suggests any doubt as to the sanctions by which they believe in the equality of the three angles of a triangle to two right angles. Let them know now that this proposition has never been proved, and never can be proved, except by the somewhat illogical process of first assuming what is equivalent to the same thing, as Euclid does in assuming the twelfth axiom. Let it be granted that this proposition is to some very minute extent an untrue one—there is nothing we know of which shows that such a supposition is unwarrantable—no measurements that we can make with our instruments, no observations that we can make with our telescopes, no