94 



nishes a complete explanation of the " imaginary " points and lines 

 in the theory of anharmonic ratios, when viewed in relation to the 

 unit radii, as already explained. In the case of coordinate geo- 

 metry of two, and even three dimensions, the possibility of interpret- 

 ing the results of a clinant operation performed on a given unit radius 

 in a given plane, allows us to understand the whole theory of " ima- 

 ginary " intersections. The theory of scalar and clinant algebraical 

 coordinate geometry will form the subject of a future memoir. 



Proceeding to quaternions, we find their laws to be the same as 

 those of clinants while the plane remains unaltered ; but if the plane 

 is alterable, they cease to be commutative in multiplication, that re- 

 lation being replaced by one between certain related quaternions 

 called their conjugates. This makes the algebra of quaternions 

 (which is not here systematized, as being too recent) entirely different 

 from that of scalars. 



In mechanics the motion of any point is not considered absolutely 

 as in dynamical geometry, but relatively to some external, constant, 

 independent motion, as the apparent motion of the fixed stars ; this 

 gives the conception of time. But the necessity of considering the 

 motion not merely of a point, but of a body, gives rise to the com- 

 parison of the motions of various bodies, and to a conception of their 

 equality, when the products of their velocities, multiplied by a con- 

 stant which is always the same for the same body, but different for 

 different bodies, are equal. This constant is the mass, which in 

 bodies of the same kind varies as the volume. 



By considering the case of the mutual destruction of motion, we 

 eliminate time and simplify the problem, thus founding statics; 

 and by conceiving the motion of any body to be destroyed by the 

 application of variable motions equal and opposite to those actually 

 existent, we reduce dynamics to statics. 



