38 Prof. Miller on the employment of the Gnomonic 



the points when their symbols are given, depending upon the 

 equality of the anharmonic ratios of points and great cii'cles on 

 the sphere, with those of the points and straight lines into which 

 they arc projected gnomonically, which I now proceed to inves- 

 tigate. 



Arts. (2), (13), though to be found in every treatise on modern 

 geometry, and art. (3), which occurs in Mulcahy's * Principles of 

 Modern Geometry,' are given in order to save the trouble of 

 reference. The results of (6), (7), (9), (10) have also appeared 

 before; the methods, however, of deducing them by spherical 

 trigonometry, now given, are very much more simple and direct 

 than those which I had previously employed. 



2. Let four straight lines passing through the point K, meet 

 any other straight line in the points A, B, C, D. 



AB sin ABK=KA sin AKB, 

 DBsinDBK = KDsinDKB, 

 AC sin ACK = KA sin AKC, 

 DC sin DCK = KD sin DKC. 

 Whence we easily obtain 



AB DC _ sin AKB sin DKC 

 DBAC~sinDKBsinAKC" " " ' 



{«) 



If the straight lines KA, KB, KC, KD meet any other straight 

 line in the points E, F, G, H, it is evident that 



ABDC_EPHG 



DBAC "HFEG ^^^ 



3. Let four great circles passing through the point K, meet 

 any other great circle in the points A, B, C, D. 



sin AB sin ABK= sin KA sin AKB, 

 sin DBsin DBK= sin KDsin DKB, 

 sin AC sin ACK= sin KA sin AKC, 

 sinDCsinDCK= sin KDsin DKC. 



Whence 



sin AB sin DC _ sin AKB sin DKC 

 sinDB sin AC ~ sin DKD sin AKC* 



(7) 



If the great circles KA, KB, KC, KD meet any other great 

 circle in the points E, F, G, H, it is evident that 



sin AB sin DC _ sinEF sin HG ^. 



siiTDB sin AC ~ sin HF sinEG ^ ^ 



