80 Improvements in Fundamental Ideas 



actual direction IA until it arrives in the actual direction 

 IB. 



If AA and BB be two straight lines, and I their point 

 of intersection, the angle ' right AB ' means the angle IA 

 right round to IB, and the angle ' left AB ' means the angle 

 I A left round to IB. 



The angle (AB) means either one or the other of these 

 last two. 



Similar remarks, as far as regards right or left formation, 

 apply to the arc of any curve : the elements of the arc being 

 supposed described by their respective radii of curvature. 



THEOREMS. 



Magnitudes of like formations, whether lines or angles, 

 should receive like signs, viz. : — 



All magnitudes of right formations should receive like 

 signs, and all magnitudes of left formations should be dis- 

 tinguished by other like signs. 



Magnitudes of opposite formations should receive opposite 

 signs, viz. : — 



Right formations should receive opposite signs to left 

 formations. 



Magnitudes of the same ' relative' formations, such as f left 

 oblique,' should receive like signs. 



Magnitudes of opposite ' relative 5 formations, such as 

 c right oblique' and ' left oblique,' should receive opposite 

 signs. 



Numbers, lines, or other magnitudes, whose ratios truly 

 express the ratios of magnitudes of like or unlike formations, 

 must accordingly be of like and unlike signs. 



If two straight lines MM MN intersect each other, then 

 the following relations exist amongst the angles at their 

 point of intersection : — 



The angles MM right to NN are equal to each other. 



The angles NN right to MM are equal to each other. 



The angles MM left to NN are equal to each other. 



The angles NN right to MM are equal to each other. 



The sum of the two angles MM right to NN, and NN 

 right to MM = half revolution right. 



The sum of the two angles MM left to NN, and NN left to 

 MM = half revolution left. 



If we have any number of points, and that we assume any 

 one of them as a starting-point, then, in respect to any pri- 

 mitive direction whatever, will the sum of the relative dis- 



