Mr J. Sang on a Systematic Method of 



grees and minutes all round the circle, and giving the signs of 

 the functions. 



Conceiving the figure to represent a series of surveying-lines, 

 whose bearings and inclinations have been ascertained in the 

 field, and whose lengths have been necessarily measured for 

 the sake of offsets, and that we assume co-ordinates for A, and 

 calculate from it the co-ordinates of all the points in succes- 

 sion until we again arrive at A, we will have before us a clear 

 view of the accuracy of the field operations. For it is appa- 

 rent that the co-ordinates of A thus calculated should be the 

 same as those assumed at first. Or, which is the same thing, 

 the sum of all the differences between each two points of alti- 

 tude, latitude, and longitude respectively, should amount to 

 nothing. If these sums be much more, or less than nothing, 

 there has been a blunder in the survey, for which there is no 

 other remedy than to revise the field operations ; but if there 

 should be a slight excess or defect from zero, we may safely 

 assume that it is composed of an accumulation of minute er- 

 rors in the co-ordinates of each point, contracted in the mea- 

 surement of the lines and bearings, from the unavoidable causes 

 already mentioned. And from this assumption it follows, that 

 a nearer approximation to the truth would be the calculated 

 co-ordinates collated with a proportion of the error depending 

 on the distances of the points from one another in order, and 

 the functions of the respective bearings of the lines. Such a 

 subdivision of error would, in a practical point of view, produce 

 quantities differing very little from the quantity given by di- 

 viding the whole error by the number of points. If we there- 

 fore divide the error by the number of points, and apply one 

 part to each, we will arrive at a very near approximation to 

 their true places ; at any rate, we will effectually prevent those 

 accumulations of error so troublesome in the old methods. By 

 using the same process in every one of the polygons of which 

 a survey consist, we will thus either have the errors entirely 

 corrected, or minutely subdivided among all the points, accord- 

 ing as our assumption applies or not to the circumstances, in 

 either' case, by a simple operation, rendering the whole survey 

 infinitely more accurate than one conducted with equal care in 

 the old method. 



