116 



LECTURES 



before any absolute distance was determined. The periodic time of the 

 several planets was very easily and very early known. And Kepler's 

 Third Law at once determined all the relative distances. Taking the 

 earth's distance from the sun as the unit of measure, v/e readily obtain 

 all the others in terms of this by the following proportion :_ 



F^: p^ : : 1 : d^ ; in which P represents the periodic time of the 

 earth, p that of the planet, and d the distance of the planet^ which 

 is the only unknown term. 



The radius of the earth being known, the absolute distance of a 

 body is determined by its parallax. This depends upon the solution 

 of a plain triangle, and, in point of principle, the problem is precisely 

 the same as that of determining the distance across a river by means 

 of measurements made on one of its banks. Thus, in fig. 11, 



*i+ 



E M', in which the first three terms 



if we suppose E to be the earth, M' the moon in the horizon as seen 

 from the station 0, then M' E will be the horizontal parallax. 

 Now, if this parallax is known, which is simply the semi-diameter of 

 the earth seen from the moon, then knowing E the radius of the 

 earth, and the right angle E M', we have in the triangle E M' all 

 the angles and one side to find E M' the distance of the moon, which 

 is done by the proportion : 



Sin. E M' 0: radius:: E 

 are known. 



Again : Suppose the moon to be at M, let and 0' be two stations 

 in the same meridian, whose latitudes are known, which makes known 

 the angle at the centre E 0'. The object is to find the distance 

 E M. To do this, we want the observed zenith distances Z M and 

 Z' 0' M, which will make known their supplements E M and E 0' 

 M ; then in the quadrilateral figure E M 0', having already three 

 of the angles, we can easily obtain the fourth M 0', by subtracting 

 the sum of the three, which are known, from four right angles. With 

 all the angles and the two sides E and E 0' it is a simple trigno- 

 metrical problem to compute the distance E M. The radii of the 

 earth being known, the only error in the result will depend upon the 

 determination of the latitudes and the zenith distances. But, with 

 the instrumental means now at command, these errors are extremely 

 small. The angle M 0' may also be determined by observed distances 



