*THE POPULAR SCIENCE MONTHLY.*

practical meaning is only very close approximation; *how* close, depends upon the circumstances. The knowledge, then, of an exact law in the theoretical sense would be equivalent to an infinite observation. I do not say that such knowledge is impossible to man; but I do say that it would be absolutely different in kind from any knowledge that we possess at present.

I shall be told, no doubt, that we do possess a great deal of knowledge of this kind, in the form of geometry and mechanics; and that it is just the example of these sciences that has led men to look for exactness in other quarters. If this had been said to me in the last century, I should not have known what to reply. But it happens that about the beginning of the present century the foundations of geometry were criticised independently by two mathematicians, Lobatschewsky^{[1]} and the immortal Gauss;^{[2]} whose results have been extended and generalized more recently by Riemann^{[3]} and Helmholtz.^{[4]} And the conclusion to which these investigations lead is that although the assumptions which were very properly made by the ancient geometers are practically exact—that is to say, more exact than experiment can be—for such finite things as we have to deal with, and such portions of space as we can reach; yet the truth of them for very much larger things, or very much smaller things, or parts of space which are at present beyond our reach, is a matter to be decided by experiment, when its powers are considerably increased. I want to make as clear as possible the real state of this question at present, because it is often supposed to be a question of words or metaphysics, whereas it is a very distinct and simple question of fact. I am supposed to know, then, that the three angles of a rectilinear triangle are exactly equal to two right angles. Now, suppose that three points are taken in space, distant from one another as far as the sun is from α Centauri, and that the shortest distances between these points are drawn so as to form a triangle. And suppose the angles of this triangle to be very accurately measured and added together; this can at present be done so accurately that the error shall certainly be less than one minute, less therefore than the five-thousandth part of a right angle. Then I do not know that this sum would differ at all from two right angles; but also I do not know that the difference would be less than ten degrees, or the ninth part of a right angle.^{[5]} And I have reasons for not knowing.

This example is exceedingly important as showing the connection

- ↑ "
*Geometrische Untersuchungen zur Theorie der Parallellinien*," Berlin, 1840. Translated by Hoüel, Gauthier-Villars, 1866. - ↑ Letter to Schumacher, November 28, 1846 (refers to 1792).
- ↑ "
*Ueber die Hypothesen welche der Geometrie zu Grunde liegen*," Göttingen Abhandl., 1866-'67. Translated by Hoüel in*Annali di Matematica*, Milan, vol. iii. - ↑ "The Axioms of Geometry," Academy, vol. i., p. 128 (a popular exposition).
- ↑ Assuming that parallax observations prove the deviation less than half a second for a triangle whose vertex is at the star and base a diameter of the earth's orbit.