75 
$ 28. To get a good insight in the significance of the field of 
ratio it is important to name some more theorems out of rational 
planimetry. 
The rational area of triangle P,,P,,P, is given in: 
1 _ dix 2 dix 2 … dl 
P (mia). P(m, a's): Pm a). 
3 1 3 
After a few reductions we find the following symmetrical form: 
2 3 I a, a, a, 
Bo IE en Ys ee (2 “1 Wenn | 
l ) 3 Vs v, vs 
In case P, coincides with M (1,1) the form becomes: 
(ys 4 2) : (421 %)- 
If three points lie on one rational, then its value becomes one, 
as is easy to see. 
For the rational area P, of the rational parallellogram (fig. 10) 
holds : 
CER AID RAZ AR Jak pod 4 2 
Pee Ne Pe eN Pea ee) 
BA a4 ket eas eee’ 3 1 
1 
in which for short for the multiplicals one letter is taken, e.g. P 
3 
for area (P,, P,, z,,7,). Out of the equidistance of the sides follows 
immediately : 
Lt Vy hy ity 5 Ya: Ya Yn YG 
so that two opposite rational sides are equal: 
2 av 2 y 2 a g y 
ns =) (#)= (=) (Ere x, 
ue te ie ER AT dae 
The analogon of the theorem of Pytsacoras can be deduced in 
the simplest way out of the equation of the logarithmic circle. For 
a rational rectangular triangle placed arbitrarily we have but to 
apply revolution about the axis (see § 15). Thus we find also easily 
the rational area of a triangle, which is the root out of the mutual 
power of a rational side and the rational height let down out of 
the third vertex on to it. The considerations of the rational vector- 
analysis lead in a shorter manner than the ways indicated here to 
the results required. 
A word or two must still be said about polar coordinates. The 
equation of the rational becomes : 
u 
Q, cr — = Os 
Uo 
when g is the rational distance M (1,1) to the point of the line and 
