ANNIVERSARY ADDRESS OF THE PRESIDENT. XXXIX 



the necessity for assuming the position of the centre of reduction of 

 the great circles of reference, and the above example shows the 

 serious error to which such an assumption may possibly lead. 



An error of this kind, however, in the provisional determination of 

 the great circles of reference, is not of material importance with 

 respect to this first part of M. de Beaumont's theory, because it does 

 not affect the question of the parallelism of systems of lines of con- 

 temporaneous origin ; but it appears to me to be of vital importance 

 with reference to the evidence which can possibly be at present 

 adduced in support of the second part of our author's theory, because 

 that evidence depends essentially, as we shall see hereafter, on a coin- 

 cidence of the great circles of reference with certain lines of his 

 reseau pentagonal, too approximate to admit of the degree of uncer- 

 tainty above- mentioned in the positions of those great circles. But 

 I shall again return to this point. 



Proceeding at present, however, on this assumption, we have only 

 to fix on a point as centrically situated as possible with reference to 

 our given lines of elevation, and to consider that point as the centre of 

 reduction through which the required great circle of reference must 

 pass. Its position must be defined by its latitude and longitude. 

 The next step is to determine the direction in which the great circle 

 of reference passes through the centre of reduction. For this pur- 

 pose we have to determine the angle which this great circle must make 

 with the meridian at that point (or the point of the compass to which 

 it is there directed) in order that it may be parallel to any given line 

 of elevation of the proposed system, the position of that line being 

 given by means of the latitude and longitude of its middle point, and 

 the point of the compass to which it is directed. If all the lines 

 were exactly parallel, in the sense in which the term is here used, it 

 would manifestly be immaterial which we should take for the purpose 

 of determining the direction of the above great circle, since each 

 would necessarily give the same result ; but in nature this parallelism 

 can only be expected to be approximate. Consequently we must de- 

 termine the directions of the great circle of reference corresponding 

 to each particular hne. These directions will be only approximately 

 and not accurately the same, and the mean of these directions must 

 be taken, as that which will best represent the great circle of the 

 system. 



The different lines of a system characterized by this kind of paral- 

 lelism, and comprised within a portion of the earth's surface suffi- 

 ciently small and not too near either of its poles, will be directed 

 sensibly to the same point of the compass ; but a little consideration 

 will show us, that such is not the case when the lines are contained 

 in a much larger area. The earth's equator cuts all meridians at a 

 right angle ; but any other great circle can only cut one of all the 

 terrestrial meridians at right angles, doing so at two diametrically 

 opposite points. At those points the direction of the great circle is 

 exactly east and west. It cuts all the other meridians at angles 

 differing from a right angle, and, consequently, at such points of 

 intersection its direction is not east and west. Similarlv small circles 



