77 
by way of points a figure which is rational congruent with a given 
figure. Rational congruence is of course originated by means of 
potential augmenting of the ratios. If in fig. 11 the ratio of the 
abscissae is equal to that of the ordinates we have ordinary congruence ; 
the points C and D then coincide with O. 
§ 30. In case two of the just mentioned four points coincide, 
the points are “corrational” ; the middle point is then situated mean 
proportionally. A more general relation for corrational points is 
Pesa = RED of (At BF (Rs OP; 
R divides the rational distance P,Q logarithmical proportionally 
according to: 
J Ys 25 Ve 
Ln Hon Q 
By drawing the root the logarithms of the rational weights (a, 5 
and a+) can be varied so that a+6=1. In fig. 11 the points 
P form the vertices of a rational parallelogram of which P, satis- 
fying: 
ber eo he Fe 
is the centre; this point is the geometrical mean of the diagonals. 
If now the point ratio is cailed the “freerational vector”, then the 
rational distance P,, P, = P,“?: must be regarded as “bound rational 
vector” (§ 27). 
In the field of difference a point ratio has no significance, the 
product of points only when the exponents are missing. It will 
therefore be right to furnish the rationai product with a multipli- 
cative sign. By ascending to the rootfield the mutual root of two 
points, having no importance for the field of ratio, will represent a 
free vector. The product of two free vectors is again a free vector; 
this can then be regarded as a resultant of the two. This is easy 
to see when we move one of the vectors until one end coincides 
with one of the ends of the other vector. 
§ 31. It is easy to see the following theorems. 
II. The product of point and free vector is a point. 
Il]. The mutual power of a point with a free vector is a bound 
vector: 
5 
P,,—=P,,P,:P,,P,=P,,P,. 
if 
: ia 
That for three points of a rational holds simultaneously : 
Br P 
mand Pk 
