and Loci of Apollonkis, ^c. 27 



revolvin.^ from a position in AA hy a right rotation until its 

 first arrival into the position BB, the revolving line being- 

 supposed produced indefinitely on both sides of the pivot. 

 And a similar meaning applies to the term ''angle lA left 

 toB." 



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

 of intersection, then " a)7ff/e I A right round to IB " means the 

 angle formed at I by a straight line having one of its ex- 

 tremities in this point, revolved by right rotation from the 

 actual direction lA until it arrives in the actual direction IB. 

 And a similar meaning applies to the term " angle lA left 

 round to IB." 



4. If A A and BB be two straight lines, and I their point 

 of intersection, the angle "right AB" means the angle lA 

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

 I A left round to IB. 



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

 last two, indifferently. 



NOTE. 



It is necessary to restrict the meaning of the term " angle 

 (AB),'' given in Chasles' Geometric Superieure, to that which 

 has been just defined ; for otherwise, his enunciated pro- 

 perties of the homographic pencils will not hold good as to 

 sign. 



See my paper entitled " Iinprovements in Fundamental 

 Ideas and Elementary Theorems of Geometry,'' in the Trans- 

 actions ^ov 1859. 



SECTION OF RATIO. 



Given the points P and R in. the given straight lines MM 

 and NN ; through a given point A to draw a straight line 

 CAD to cut the given lines in C and D, so that the segments 



