ANNIVERSARY ADDRESS OF THE PRESIDENT. XXXVU 



the middle and westerly parts of Europe, and let us suppose the 

 determmation of the direction of each line of the system, subject to 

 an error of 3"^, i.e. of 1^° in excess or defect, the consequent error 

 to which the determination of the pole of the system, by means of any 

 two of these lines, would be liable, might amount to 8° or 1 0°, though 

 we should select two lines best adapted for this determination ; and 

 if we also take account of the actual deviations of the existing lines, 

 from the exact parallelism above assumed, the above error might be 

 much greater still. The pr-obable errors would doubtless lie within 

 these extreme limits ; but even these errors must necessarily be very 

 considerable, when the determination of the position of the pole of 

 the system, or of the great circle of reference, depends on lines 

 spread over no larger a portion of the earth's surface than that of 

 Western and Middle Europe. This, I imagine, is the reason why 

 M. de Beaumont has not attempted to determine, by the above or 

 any equivalent method, the positions of the great circles of refer- 

 ence for his European systems. Guided by certain hypothetical 

 considerations, as I shall presently show, he has in each case 

 assumed a point through which each of these circles passes, and 

 regards their positions, as at present determined, onij provisional, 

 to be corrected hereafter when all the component lines of each 

 system shall be more accurately known. 



In all the preceding explanations I have made much more use than 

 M. de Beaumont of the s?nall circles parallel to each great circle of 

 the sphere, and of their pole, believing that I might thus facilitate 

 your accurate conception of the leading points of this theory. Our 

 author, on the contrary, usually appeals to the great circle of refer- 

 ence itself rather than to its pole. The position of that circle may 

 be determined by the latitude and longitude of its pole ; or by the 

 position of a point called the centre of reduction, through which the 

 circle passes, and the angle which it makes with the meridian at that 

 point, i. e. the point of the compass to which it is there directed. It 

 is this latter method which M. de Beaumont has found it most con- 

 venient to adopt, and those who may follow his steps will find the 

 facilities of doing so very much increased by certain tables which he 

 has given for the purpose. And, indeed, we cannot too highly com- 

 mend the pains which he has taken throughout to secure the accuracy 

 of calculations involving an immense mass of details, and to aiford 

 every facility either for the verification of his own results, or for the 

 making of other similar calculations. 



Our author has given strictly accurate methods of determining the 

 two elements which fix the position of the great circle of reference of 

 any proposed system of lines, viz. the centre of reduction, and the 

 direction in which the great circle passes through that centre. These 

 methods, however, involve very complicated and laborious calcvdations, 

 into which it would be useless to enter without more accurate data 

 than we at present possess. The methods which he has actually 

 adopted are such as lead to results of approximate accuracy. An 

 exact calculation with imperfect data would only lead to results 



