Mr. A. J. Ellis on the Systematization of Mathematics. 231 
of four dimensions with clinant coefficients, and to show that every 
formation consisting of a sum of integral powers with clinant coeffi- 
cients, can be expressed as a product of as many simple formations as 
is determined by the highest index of the variable. The cosine and 
sine series can also be generally inverted. The versor of any clinant 
having a known angle (which is aiways equal to the cosine of its 
angle added to the product of the sine of its angle into a quadrantal 
yersor), can now be shown to equal the cosine series added to the sine 
series multiplied by a quadrantal versor, when the variable of the 
series is the scalar ratio of the angle of the clinant to the angle sub- 
tended by a circular are equal to its radius. From this the ratio of 
the circumference to the diameter of a circle is shown to be twice the 
least tensor value of the variable, for which the cosine series is equal 
to null; and as that value can be readily assigned in a convergent 
series, the former ratio is determined. The same investigation shows 
the relation already mentioned between the goniometrical cosines 
and sines, and the cosine and sine series. 
Clinant algebraical geometry allows us to interpret all results of 
clinant algebra when referred to lines on one plane. It thus fur- 
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. 
