XXXVIU PROCEEDINGS OF THE GEOLOGICAL SOCIETY, 



having the same degree of iudeterminateness as that of the graphical 

 method above explained of determining the pole of the proposed 

 system, so far as that iudeterminateness depends on the imperfect- 

 ness of the data. Instead, therefore, of calculating for each system 

 the position of the centre of reduction through which its great 

 circle of reference passes, our author has, in each of the systems 

 which he has investigated, assumed the position of this centre, guided 

 by such considerations as the following : — If the space occupied by a 

 system of parallel lines on the earth's surface were a well-defined 

 zone bounded by two parallel small circles, to which all the lines of 

 the system are parallel, the great circle of reference must neces- 

 sarily be that which is parallel to these bounding small circles. It 

 might be vdthin or without the zone. If the zone were of considerable 

 length, there would be no difficulty in determining its great circle, 

 either by calculation or by the graphical method ; but when the 

 given part of the zone is very short (not extending, for instance, beyond 

 Western and Middle Europe) and only approximately known, it will 

 be impossible to determine the exact position of the complete sone, or 

 consequently to find, with any very approximate accuracy, the 

 position of its great circle of reference. It is here that our author 

 introduces his assumption — that the sone, in lohich each actual 

 system of parallel lines of elevation on the earth's surface is com- 

 prised, is bisected, or nearly so, by its great circle of reference. 

 Whether this assumption be true or not, it is impossible to deter- 

 mine, till the systems recognized by M. de Beaumont have been 

 traced over a much greater extent of the earth's surface. There is 

 probably scarcely one of the European systems in which its great 

 circle of reference would not be as good an approximate represen- 

 tative of the direction of the system, as at present, if the centre of 

 reduction, vnth. the circle itself, were moved 8° or 10° from its pre- 

 sent position in a direction perpendicular to that of the provisional 

 circle of reference. 



The great uncertainty which attaches to this method of deter- 

 mining the centre of reduction from observations in a limited district, 

 may be elucidated by considerations like the following. Suppose that 

 instead of taking the region of Western and Middle Europe, we should 

 take that of Eastern Europe and Western Asia. Proceeding on the 

 above assumption respecting the centre of reduction of each great 

 circle of reference, most of those great circles would be made to pass 

 nearly centrally through this new region, instead of being so situated 

 with reference to the former region. This would be a manifest con- 

 tradiction, supposing always the same parallel systems to exist in 

 both regions. Either of these determinations might be right, one of 

 them must be wrong, and probably both would be so. If the direc 

 tions of the lines of elevation could be accurately observed, and were 

 accurately parallel, there could be no such contradiction as the above. 

 It is the want of this geometrical accuracy, and the small dimensions 

 of the observed region compared with the area of the complete ter- 

 restrial zone, of which it may be considered a portion, which create 



