ANNIVERSARY ADDRESS OF THE PRESIDENT. XXIX 



In considering the nature and relative positions of lines drawn on 

 the surface of a sphere, we may regard them as straight or curved, 

 as parallel or inclined to each other at certain angles, in the same 

 sense in which we should speak of such lines drawn on a surface ac- 

 curately plane, provided the extent of the surface on which these 

 lines are situated be so small compared with the surface of the whole 

 sphere, that we may neglect the curvature of this small portion of 

 the spherical surface without committing an error large enough to 

 be of importance in the final results of our investigations. Thus, in 

 speaking of the parallelism of any phsenomena of elevation in a par- 

 ticular geological district, it would only be an absurd affectation of 

 accuracy to regard such lines as drawn on the spherical surface of 

 the earth in contradistinction to their being drawn on a plane surface, 

 though it might be absolutely necessary to make the distinction with 

 reference to the refined exactness of a trigonometrical survey. In 

 such limited regions the term parallelism may receive its ordinary 

 definition without sensibly affecting any degree of accuracy to which 

 geological investigation or observation can aspire. But, on the con- 

 trary, when we speak of lines on the earth's surface distant from 

 each other by many degrees of latitude and longitude, we can no 

 longer, even in an approximate sense, consider them as straight lines 

 drawn on a plane surface, and consequently the term parallelism, 

 which, in its strict definition, is applicable only to straight lines, can 

 no longer be applied except under some modified definition of the 

 expression. It is absolutely necessary to understand the exact sense 

 in which M. Elie de Beaumont makes use of this term. To explain 

 it, I shall first give the definitions of a few elementary terms of 

 constant occurrence in the geometry of the sphere. In doing this I 

 may appear perhaps to be assuming an ignorance on points with 

 which the majority of my hearers are well acquainted ; but I think 

 it better not to incur the risk of using terms in any part of my ex- 

 position of this theory, which might convey erroneous, or at least 

 indefinite conceptions to the mind of any geologist who may wish to 

 make himself acquainted with it. 



Suppose we make a number of sections of a sphere by a set of 

 planes all parallel to each other, and one of which passes through 

 the centre of the sphere. The section of the surface of the sphere 

 made by this latter plane will be a circle which will manifestly divide 

 the whole surface of the sphere into two equal portions. Such a 

 circle is termed a great circle of the sphere. The sections of the 

 surface of the sphere formed by the other planes of the set above- 

 mentioned will also be circles, but such as will divide the whole sur- 

 face of the sphere into two imequal portions. All such circles are 

 called small circles. Thus we have a set or system of small circles, 

 indefinite in number, corresponding to each great circle, and all 

 having a certain kind of parallelism to each other depending on the 

 fact of the planes by which they are formed being |jGr«Z/e/ to each 

 other in the strict geometrical acceptation of the term. It is on this 

 kind of parallelism of one of these circles to another, that the defi- 

 nition of the term, as used by M. Elie de Beaumont, is founded. 



