118 



LECTURES 



planet Mars in opposition, contemporaneously with Cassini in Paris, 

 Wargentin in Stockholm, and others in the north of Europe. These 

 observations gave for the parallax of the sun 10". 4 nearly. The 

 parallax of Mars was about 25". 



The method by which the results were obtained will be understood 



Fig. 12 



by a reference to figure 12. S on the left is the position of the sun, 

 E the earth, M Mars in opposition. The stations of Wargentin at 

 Stockholm and La Caille at the Cape of Good Hope were nearly on 

 the same meridian. Let them be represented by and 0', their 

 latitudes being; known. Then in the quadrilateral figure E 0' M, 

 the angle at E is known by the latitudes ; the angles at and 0' are 

 known by the observed zenith distances Z M and Z' 0' M. These 

 three angles would make known the angle at M. But to avoid the 

 possible errors in determining the first three it is better to obtain 

 M by direct observation. Its value will serve as a test of the cor- 

 rectness of the other angles. This angle at M is most accurately 

 found by measuring the apparent distance, at each station, be- 

 tween Mars and some star near it which passes the meridian at 

 nearly the same time. At this distance would be S M; at 0' 

 it would be S' 0' M. The angle at M is the difference between these 

 two. For, on account of the great distance of the star, S and 0' 8' 

 are parallel ; and therefore S M = X 0'; but this latter being 

 exterior to the triangle 0' X M, we have X M 0' = X 0'— X 0' M ; 

 hence the angle at M is equal to the difference between the observed 

 distances of the star from Mars. The radii of the earth E and E 0' 

 being known, it is easy to compute the distance E M, the distance of 

 Mars from the earth. It will now be very easy to find E S, the distance 

 of the sun from the earth. From Kepler's Third Law we know that the 

 relative distances of the earth and Mars from the sun are as 100: 152 

 nearly ; that is, if the earth's distance were divided into 100 equal 

 parts the distance from the sun to Mars would be 152 of those parts, 

 and the distance from the earth to Mars in opposition would be 52 of 

 the same parts. But the foregoing process gives us the absolute value of 

 those 52 parts, and hence we can by a simple proportion find the value 

 of the 100 parts between the earth and sun. Let m .= the number of 

 miles in E M; then to find the number in E S, we have 52 : 100 : : m: 

 ES; orES= ^ 



The observations of La Caille and Wargentin were made on the 6th 

 of October, 1Y51. They gave for the parallax of Mars 24". 7, for that 

 of the sun 10". 4, and for the distance of the sun a little more than 



