OF ARTS AND SCIENCES. 261 



since the elevations of all other mountains in sight, except those in the 

 immediate vicinity, may be determined. 



The fact is worth noting, that, even if the error from refraction 

 could be eliminated, the method of zenith-distances would not equal in 

 accuracy that of ordinary levelling. For suppose that the sights are 

 taken at distances c?, and that the probable error of each is e. If n 

 sights are taken, or the distance travelled is nd, the probable error 

 will be only e^n, since the positive and negative errors will probably 

 in part neutralize. The error in the method of zenith-distances will, 

 however, be proportional to the distance, or will be ne. Thus, if sights 

 are taken every hundred metres with a probable error of 1 mm., the 

 probable error of the level in ten kilometres will be 10 mms., since 

 n = 100; while with zenith-distances the error would be 100 mms. 

 Evidently, therefore, within reasonable limits, to attain the greatest 

 accuracy with the level, the sights should be as short as possible, — a 

 fact in accordance with general experience. 



Evidently the telescope of a theodolite may be converted into a 

 micrometer level by inserting in its eye-piece a scale, and attaching 

 three levels. Small vertical angles, which are those most used in 

 surveying, can then be measured more accurately than by a vertical 

 circle. In the same way, a similar attachment may be made to the 

 telescope of a plane-table ; and the advantage is especially marked in 

 this case, since an accurate mounting is not needed. A further appli- 

 cation may then be made ; namely, to determine the distance when the 

 height is known. Suppose that a distant pond is observed from the top of 

 a hill. The telescope is directed to various portions of its shore, and 

 the apparent depression observed. Since every portion must be at an 

 equal distance below the observer, it is easily shown that the distance 

 is always inversely proportional to the depression. Accordingly, if 

 the direction of each part of the shore is marked on the plane-table 

 sheet by a line, and a distance is laid off on this invei'sely as the 

 observed depression, a map of the pond is quickly made. This may 

 be reduced to its true scale if we know the position of any one point, 

 or the height of the observer above the water. If the shore is abrupt 

 or wooded, only the farther edge of the shore can be thus surveyed, or 

 rather the portion where the actual shore-line is visible. Tliis method 

 is in fact a form of stadia, in which the measuring-pole is replaced l)y 

 the constant vertical distance between the eye and the plane of the water. 

 The same method may be used for determining the form of an island, 

 of a coast-line, or of a river winding through a nearly level meadow. 

 The form of a pond or island may also be obtained in the same way 



