and Loci of Apollonius, S^c. 25 



tlieoreras) by means of whiclitlic conclusion is drawn from those 

 premises, arc true, unambiguous, and correctly understood. 



This being borne in mind, it is evident that if from a given 

 point in a given indefinite straight line, avc were told to cut off 

 a part equal in length to a given finite straight line, we should 

 naturally ask in which direction from the point we are to 

 take the part; as the problem would be ambiguous if either 

 direction should not be answerable to the end in view. And 

 if the solution of some other problem depended on this opera- 

 tion (as just defined), and that one only of the parts Avhicli 

 can be cut off is applicable, then it is evident there would exist 

 an ambiguity in the solution. 



The method of indicating opposite directions on the same 

 straight line in distinct terras — such as positive and negative 

 directions, or right and left directions, obviates this difficulty; 

 but though long since adopted in Trigonometry and Algebraic 

 Geometry, it is only in the modern French pure geometry, 

 that it has been consistently introduced. 



Again, if in any general investigation or construction it were 

 necessary to draw a straight line through a particular point, 

 making an angle of a given magnitude Avith another straight 

 line, then, as two such lines can be drawn through the point, 

 and that but one of them may be answerable, it is clear there 

 should be a precise method of indicating each of these lines. 



Further, if on any straight line, for instance, a particu- 

 larised one of the last two, it were necessary to cut off a segment 

 from a point therein Avhose length should have some peculiar 

 relation to other magnitudes and positions, and that but one 

 segment from the point would be answerable ; then, too, it is 

 obvious we should have a method of particularising directions 

 in one straight line in respect to the directions in others. 



Yet it is only in my previous papers such methods are 

 either advocated or applied. 



And without those improvements in the manner of indicat- 

 ing angles, it is not only the elementary geometry of the 

 straight line and circle that suffers, but also the conic sections 

 and higher departments; for there, too, geometers have failed 

 to expose the general tniths comprehended in the theory. 

 One instance of this is supplied by the following wcU-knoAvn 

 theorem : — " When the base AB of a triangle is given in 

 position and magnitude^ and that the difference of the angles 

 CAB, CBA at the base is constant, then will the locus of the 

 vertex C be a hyperbola." 



For, as the locus of the vertex is not a hyperljola under 



