XXX PROCEEDINGS OF THE GEOLOGICAL SOCIETY. 



As the centre of a circle is equidistant from every point in its cir- 

 cumference, so is there a point on the surface of the spliere equi- 

 distant from every point of a great circle of the sphere. There are, 

 in fact, two such points at opposite extremities of that diameter of 

 the sphere which is perpendicular to the plane of the great circle. 

 Each of these points is also equidistant from every point in the cir- 

 cumference of any small circle, whose plane is parallel to that of the 

 great circle. These two diametrically opposite points are called the 

 poles of the great chcle, from every point of which they are equi- 

 distant. They may in like manner be termed the poles of all the 

 correspondmg small circles, the only difference being that the same 

 small circle is not, as in the case of a great circle, equidistant from 

 each of its poles. The equator and the small circles parallel to it, 

 as represented on a common terrestrial globe, offer an example, fami- 

 liar to every one, of a system of circles like that I have described, 

 the poles of the earth being in this case the poles of the system. 



We may observe that the position of the pole of a great circle 

 determines that of the great circle itself, as the pole of the earth 

 determines the position of the equator. The position of a system of 

 small circles is detemiined by that of its great circle, or, therefore, 

 by that of its pole. The position of each individual small circle is 

 not completely determined by that of its pole, since its distance from 

 the pole remains undetermined. 



If Ave draw a great circle through the two poles of our system, it 

 wiW. cut not only the great circle, but also every small circle of the 

 system at right angles. A number of such circles are represented 

 on common globes for the purpose of showing the longitudes of dif- 

 ferent places. Conversely (and this should be carefully noted) if we 

 draw a great circle perpendicular to a great or small circle of the 

 same system, it must necessarily pass through the pole of the system. 

 Hence we have an obvious way of finding the pole of a system when 

 any one of its circles is given ; for if we draw two great circles 

 through two different points of the given circle, oach being perpen- 

 dicular to it, then, since each must pass through the pole of the 

 system, that pole must necessarily be the point of intersection of 

 those great circles. Moreover the pole being known, the great circle 

 of the system is knovra, being that circle every point of which is 

 distant from the pole by 90° measm-ed along a great circle drawn 

 through the pole. Hence, if, at any given point of a sphere, we 

 know the direction of a small circle belonging to a particular system, 

 so as to be able to draw a great circle perpendicular to it ; and at 

 another given point, we know the direction of the same or of any 

 other small cri'cle belonging to the same system, we can at once deter- 

 mine the position of the pole and great cu'cle of the whole system ; 

 for we have only to draw two great circles respectively perpendicular 

 to the two given directions of the small circles, and the intersection 

 of these great circles determines the pole, and thence the great circle 

 of the system. A distinct conception of this simple method is essen- 

 tial to the understanding of the theory which I am about to explain. 

 If absolute accuracy were required, the position of the pole must be 



