to an independent algebra of areas, in which the generating lines are 

 considered immediately. The laws of the relations of lines thus 

 discovered, are shown to be identical with the laws of the relations 

 of tensors. Consequently, with certain limitations, the whole of the 

 algebra of tensors may be interpreted as results in the algebra of 

 areas. This leads to a perfect conception of the principle of homo- 

 nomy, or dissimilar operations having the same laws, and conse- 

 quently the same algebra. 



In dynamical or modern geometry r , all lines are considered as in 

 construction, having initial and final points. If the initial points of 

 any two straight lines are joined to a third, not on either, and the two 

 parallelograms be completed, the lines drawn from the point parallel 

 to the given lines are dynamically equal to them ; if these last 

 lie on each other, the first two lines have the same direction ; if the 

 last have only one point in common and lie in the same straight line, 

 the first have opposite directions ; and if the last do not lie in the 

 same straight line, the first have different directions, and the angle 

 between the last is the angle between the first lines. Similar defini- 

 tions can be given of direction in the case of angles and circular 

 arcs. If from the final point of any line we draw a line equal to 

 a second, and join the initial point of the first with the final point 

 of the line thus drawn, we are said to append the second to the first, 

 and the joining line is called the appense of the other two. The 

 laws of appension are shown to be the same as those of addition, 

 and are hence expressible by the same signs of combination, the 

 difference in the objects combined preventing any ambiguity. We 

 thus get the conception of a point as an annihilated line. 



The tensor operation, considered dynamically, leads to the opera- 

 tion of changing a line dynamically so that it should bear the same 

 relation to the result as two given lines bear to each other in magni- 

 tude and direction. This assumes three principal forms according 

 to the difference of direction. If there is no difference of direction, 

 the operation is purely a tensor. If the directions differ by a semi- 

 revolution, the rotation of one line into the position of the other may 

 take place on any plane. The operation is then termed a negative 

 scalar; the tensor, which includes the operation of turning through 

 any number of revolutions, is distinguished as a positive scalar. If 

 the rotation be through any angle, but always on the same plane, 



