292 



PROCEEDINGS OF SECTION A. 



shall presently endeavor to show. Referrmg again to Fig. 1 ; if it 

 were desired to represent the sun's path in any declination by a great 

 circle, that is, a circle of the same size as the equator, instead of a 

 small circle, as shown by D L, it would have to be distant from the 

 equator by r X tangent declination, as shown by the dotted line 

 D' L', instead of the sine as D L, as is also shown in Fig. 2. 



If, however, E Q, and D' L' were both projected vertically into 

 the plane of the horizon H R, it is evident that they would not be 

 distant from each other by E D, the tangent of declination, but by 

 E F. But E F is the residt of multiplying ;• tangent declination into 

 cosine latitude, .'. equal ellipses representing the sun's path on the 

 equator, and his path in any other declination when projected 

 vertically into the plane of the horizon would have their centres 

 distant from each other by the following quantity — (r tangent decli- 

 nation X cosine latitude). As, however, it is impossible to slide the 

 ellipses nlong the paper, or even to draw a separate ellipse for each 

 degree of declination, we are reduced to the expedient of supposing 

 the position of the observer to be moved in the opposite direction to 

 an equal distance, that is to say, with north declination he \\ ould 

 have to be moved south and vice versa, a clistance equal to (r tangent 

 declination X cosine latitude). As this quantity varies with the 

 latitude, a separate scale of declinations would have to be made 

 for each degree of latitude, and though the sun's true bearing in 

 any latitude and with any declination might be taken from a 

 diagram constructed on the principle of I'ig. 4, it would be a 

 comparatively complicated operation. 



Fig. 4, I may state, was the form in which my first diagram Avas 

 constructed, and, while experimenting with it, the idea occurred to 

 me that it might be possible, instead of varying the scale of 

 declination for each degree of latitude, to vary the size of the 

 ellipses in, of course, the inverse ratio. I therefore decided, instead 

 of multiplying tangent declinations by cosine latitude, to divide the 

 major and minor axes of each ellipse by that quantity (cosine 



latitude). This gives for 



Fig. 5. 



semi - major axis r secant 



latitude and for semi-minor 



sine latitude 



axis /• : — = — , — r- = r 



cosine latitude 



tangent latitude. It will 

 easily be seen that this pre- 

 serves the relative lengths 

 of the major and minor axes 

 for any degree of latitude, 

 because 1 : sine z= secant ; 

 tangent (Fig. 5). 



Here occurred a very happy 

 coincidence. In increasing 

 the size of the ellipses the 



