290 



PROCEEDINGS OF SECTION A. 



the semi-major axis of which will be equal to the radius of the 

 circle H R, and the semi-minor axis to the radius X sine Z C E, 

 which is the sine of the latitude. In the same manner it may be 

 shown that, with any latitude, if the circle representing the 

 equator be projected vertically into the plane of the horizon, its 

 projection will be an ellipse which will have its major and minor 

 axes in the proportion of 1 : sine latitude. 



I will now ask the members to imagine two extreme cases. 

 Suppose, in the first place, that an observer is situated at, say, the 

 north pole. From this point of view the sun's path would evidently 

 be a circle, which it would also be according to Fig. 1, because an 

 ellipse whose semi-minor axis is equal to semi-major axis X sine 

 90° would be a circle ; and, again, his bearing at any particular 

 time would not be affected by his declination, the altitude only 

 being altered, that is to say, his bearing would be exactly the 

 same at the same hour, say, Greenwich time, all the year round, 

 and of course with any declination. 



Suppose, again, that the observer is on the equator and the sun 

 is in declination 0, or also on the equator, it is self-evident that he 

 would rise due east, pass directly overhead, and set due west, so 

 that his path projected on the plane of the horizon would be repre- 

 sented by a straight line, which it would be according to Fig. 1, 

 because with lat. the circle E Q. would be projected edgewise. 



If the observer were still on the equator and the sun's declination 

 Avere, say, 20° N., his rising amplitude would be E. 20° N. (Fig. 2), 

 meridian zenith distance 20° N., and setting amplitude W. 20° N., 

 so that his path might still be represented by a straight line, but 

 distant from the equator by the sine of 20°. 



As the sun's path when off the equator is a small circle it would 



be represented (Fig. 2) by the line D L, which is shorter than the 



-p, c) diameter W E, and 



distant from it by r 



X sine declination. 



If, however, it were 

 required to represent 

 the sun's path in dif- 

 ferent declinations by 

 a line of constant 

 length, as D' L', which 

 is the same length as 

 W E, it would have to 

 be removed from the 

 equator by a distance 

 equal to r X the tan- 

 gent of the declination 

 in order to make the 

 rising and setting am- 

 plitudes work out as 

 by calculation, as can 



