ioo 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 knowl- 

 edge 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 exact- 

 ness 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 ; a whose results have been extended and general- 

 ized 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 practi- 

 cally 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 a 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. 6 And I have reasons for not knowing. 



This example is exceedingly important as showing the connection 



1 " Geomdrische Untermchungen zur Theorie der Parallell'mien" Berlin, 1840. Trans- 

 lated by Houel, Gauthier-Villars, 1866. 



2 Letter to Schumacher, November 28, 1846 (refers to 1792). 



3 " Ueber die HypotJiesen welche der Oeometrie zu Grunde liegev," Gottingen AbhandL, 

 1866-67. Translated by Houel in Annali di Matematica, Milan, vol. lit. 



4 " The Axioms of Geometry," Academy, vol. i., p. 128 (a popular exposition). 



6 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. 



