192 Mr. Nixon on the Measurement by Trigonometry of the 



pendicular to DC, draw through D the corresponding hori- 

 zontal line DH'. The angle formed at E or D by the meet- 

 ing of a straight line passing through any point, and a hori- 

 zontal line situated in the same vertical plane, is termed the 

 elevation or depression of that point, according as it lies 

 above or below the horizontal line: thus the angles HED, 

 H'DE are both depressions. To find by trigonometry the 

 difference of altitude of DE, no more data are required than 

 the lineal distance EE' or DD', and the depressions HED, 

 H'DE; or, simply, either of them, provided we have also 

 given the terrestrial arc E'VE, or angle E'CE. Through C 

 draw a line perpendicular to the chord of equal altitude or 

 level E'E, which line, from the properties of the circle, will 

 bisect the contained arc (at V). Make EC parallel to VC, 

 whence VCE or half the arc will be equal to CEO. As 

 HEC and E'EC are both right angles, the horizontal line 

 EH will be elevated above the chord of level E'E by the angle 

 HEE' = CEC'= half arc. It is also similarly demonstrable 

 that the horizontal line DH' must be elevated above the chord 

 of level DD' by the half arc DCV' = VCE, and that the angle 

 D'DE, as the chords DD', EE' are parallel, will be equal to 

 DEE' the angular difference of level. The depression at the 

 upper station will consequently exceed the angular difference of 

 level by the half arc, and the elevation at the lower station will 

 be equally in defect; whence the depression must exceed the 

 elevation by the contained arc, and half their sum will be equal 

 to the angle DEE'. When both angles are depressions, that 

 at the upper station (or larger one) is the half arc plus DEE', 

 and the other is the half arc minus DEE', so that their sum 

 equals that of the half arcs, and half their difference the angle 

 DEE'. With this angle, the distance EE', and the angle 

 E'DE ( = 90° plus half arc*), we get the side E'D or difference 

 of altitude. When one only of the angles is given, the dif- 

 ference of the half arc and the. depression (considered as an 

 elevation when the former exceeds the latter), or the elevation 

 increased by the half arc, will give the angular difference of 

 level f. 



Error of Collimation. — When the cylindrical rings of a 

 telescopic-level are unequal in diameter, the line of collimation, 

 supposed to be horizontal, describes, in a revolution in azi- 

 muth, the sides of a cone, erect or inverted, according as the 

 line passes below or above the true horizon to which the axis 



* In the calculations, E'DE has been considered as a right angle. 



f In latitude 54° the log. of the mean value of the half arc in seconds 

 may be found by subtracting the constant logarithm 2*308227 from the 

 log. of the distance in feet 



of 



