82 



Improvements in Fundamental Ideas 



If pp and qq be fixed straight lines, and P and Q, fixed . 

 points, and that mm and nn are fixed straight lines and M 

 and N fixed points ; if straight lines m'm' and n'n passing 

 respectively through M and N make the angles m'm' right to 

 mm, and n'n right to nn, respectively equal to the angles p'p' 

 right to pp and q'q' right to qq — where p'p' and q'q' are any 

 two straight lines through P and Q,, intersecting in the cir- 

 cumference of a fixed circle through P and Q, — then will 

 the locus of the intersection of m'm' and n'n be a fixed circle 

 passing through M and N. 



And if the angles mm right to m'm', and nn right to n'n' be 

 respectively equal to the angles p'p' right to pp and q'q' right 

 to qq ; the locus of the intersection of mm and m'm' is a fixed 

 circle passing through M and N. See " Note" at end of 

 theorems. 



If a circle always pass through two points, should its 

 radius become infinite, the straight line through these two 

 points lies in its circumference. 



Any straight line can be regarded as an arc of an infinitely 

 great circle, and if we know on which side of the line the 

 infinitely distant centre is, we then know the right and left 

 directions on the line, &c. 



If any hypotheses causes two points in a straight line or in 

 the circumference of a circle to coincide, then if these points 

 have been always or not necessarily always situated in the line 

 or circle, so accordingly will any other circle or straight line 

 through these points touch or coincide with this line or circle. 



PRACTICAL APPLICATIONS OF THE PRINCIPLES CONTAINED IN 

 THE PRECEDING PAGES. 



