Fr wki.ant).— The \o)i- I'.Kcliiliini < 1 rutin try Vindicated. 63 



point and partly on the nature of the space under consideration 

 (i.e., on the degree to which the space deviates from the pro- 

 perties of the ideal space of Euclid). For there are spaces and 

 spaces which satisfy Lobatchewsky's conditions. There is only 

 one space which satisfies Euclid's conditions, hut there is an 

 infinite number satisfying Lobatchewsky's. They vary through 

 infinite gradations, from one which has such feeble " negative 

 curvature " that it can hardly be distinguished from Euclidian 



space, to one which has such strong 

 " negative curvature " that even P Q 

 (in the annexed figure) would not meet 

 A B, but would rapidly come to its point 

 of minimum distance (M A'), and would 

 then recede for ever from A B. 

 Now, in regard to the space we actually live in, we ought, in 

 my opinion, to say this : " It may be Euclidian, or it may have 

 negative curvature : but if it has negative curvature, that curva- 

 ture must be excessively weak, though not infinitely weak, as is 

 suggested." Professor Clifford puts the case very well in his 

 lecture on "The Aims and Instruments of Scientific Thought." 

 He says : " Suppose that three points are taken in space, dis- 

 tant 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 ; hut also I do not know that 

 the difference would be less than ten decrees, or the ninth part of a 

 right angle. And I have reasons for not knowing." 



Clifford introduces this example by saying, what requires to 

 be much insisted on, that these speculations on non-Euclidian 

 space are not merely questions of words, as many people 

 imagine, but that the issue involved is " a very distinct and 

 simple question of fact." In plain language, geometry is a 

 physical and experimental science, just as much as optics or 

 physiology ; and the properties of space cannot be evolved from 

 man's inner consciousness, but must be determined by experiment 

 and observation. There was as much justification, before the 

 curvature of the earth was known, for erecting into an axiom 

 the proposition that all verticals are parallel — (For myself, I 

 cannot, even now, imagine its falsehood, although I of course 

 know it to be false) — as there is now for the statement, a priori, 

 that two shortest lines cannot enclose a space, or that the three 

 angles of a triangle are exactly equal to two right angles. 



10. " . . . it appears to me that even if the angle of con- 

 vergence is infinitely small the lines would intersect, but not, of 



