66 Transactions. — Miscellaneous, 



even though divergent at first. In Lobatchewsky's space the 

 three angles of a triangle are always together less than two right 

 angles: in the "finite plane" (and also in the corresponding 

 space of three dimensions,) they are always greater than two 

 right angles, just as the angles of a spherical triangle are. In 

 Lobatchewsky's space, figures have their edges and corners 

 sharpened when their linear dimensions are proportionately 

 increased : in the " finite plane" they have their angles blunted 

 on being magnified, (like the figures on a sphere,) and in the 

 corresponding space of three dimensions solid figures would also 

 have their edges and corners blunted on being magnified. 



17. " It is, I think, abundantly evident that the analytical 

 conception of a surface such as the one which has been worked 

 upon for the discoveries communicated in his (Mr. Frankland's) 

 paper, is not, in reality, valid, and that though possibly not 

 self-contradictory, as he urges, it requires premises which 

 are of this nature " — i.e. self-contradictory (p. 107). Not so. 

 The premises are not self-contradictory, but only contradictory 

 to some of our strongest and firmest intuitions — viz., our 

 space-intuitions. But so is the convergence of verticals, 

 already alluded to, and yet it is an unquestionable fact. 

 Believing, as I do, that our space-intuitions are derived simply 

 from ancestral experience, aided by natural selection (which 

 must always have tended to eliminate those in whom such 

 intuitions were relatively weak), I can only admit that they 

 are reliable enough for practical purposes; not that they are 

 exactly true through all space and time. The parallelism 

 of verticals was an intuition, (a sort of dynamical intuition,) 

 ingrained in our mental constitution by ancestral experience 

 through innumerable generations. Were we blind, and con- 

 fined (say by surrounding climates of excessive rigour) to a 

 very limited area of the earth's surface, I think it very likely 

 that this conception would to this day seem to us self-evidently 

 true. It would seem as certain that two verticals must have 

 the same direction as it now does that two shortest lines cannot 

 enclose a space. A Skey, in such a world, might even have 

 argued that to construct a system of cosmography in which 

 two verticals should not have the same direction would be, 

 '• though possibly not self-contradictory," to assume "premises 

 which are of that nature." In any case, I do not think that 

 any self-contradiction can be shown to be involved in the 

 proposition that two geodesic lines, though finite in length, 

 intersect only once. 



18. "Referring to the idea that the universe is of finite 

 extent," . . . the Professor " argues that ' in this case the 

 universe is again a valid conception . . . for the extent 

 of space is a finite number of cubic miles'" (p. 107). In 

 this quotation from Professor Clifford, two important words 



