10 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
0 , 
B' 
E 
Fig. 6. 
M=tt B — /?, 
we deduce from equation (3) 
P— y+z), 
which combined with equations (1) and (2) gives 
A -j- B -J- C — 7T — a — /?. 
So in Bertrand’s demonstration is already assumed a = 0, /? 
the very thing- which was to be proved. 
Just as Bertrand was satisfied with the comparison of infinite 
surfacee in the angies of the triangie, so Legendre wished to use 
the infinite surfaces between pairs of perpendiculars, which he 
called biangles. 
Really he only demonstrated that the infinite 
surface CABD (Fig-. 6), between the perpendic- 
ulars AC, BD, to AB is equal to the infinite sur- 
face DEFC obtained from the preceding- in cut- 
ting- off the quadrilateral ABEF by the perpen- 
dicular EF to BD. This is sufficiently evident; but Leg-endre has 
not noticed here, that EF may possibly not meet AC. 
To overcome this little difficulty, you have only to suppose that 
EF is the perpendicular from F on BD; but then how can we con- 
clude therefrom that FE=AB and the ang-le EFC=-£ar? It is not 
possible to mend the false deduction, wherein Leg-endre ’s inad- 
vertence was so gross that without remarking- this grave error, 
he considered his demonstration as very simple and perfectly rig- 
orous. 
Again the plan has been thought of in the theory of parallels to 
use as foundation, that in triangles the angles must depend on the 
ratios of the sides. 
At first such an assumption seems as simple as necessary; but 
if we seek to probe what idea we have thereof, whence it takes 
its source, we are forced to designate it as just as arbitrary as all 
others to which recourse has been had. 
We cognize directly in nature only motion, without which sense- 
impressions are not possible. Consequently all other ideas, for 
example geometric, are artificial products of our mind, since 
they are taken from the properties of motion; and consequently 
space in itself, for itself alone, for us does not exist. Accordingly 
it can have nothing contradictory for our mind, if we admit that 
some forces in nature follow the one, others another special 
geometry. 
To illustrate this thought, assume, as many believe, that attract- 
ive forces diminish because their action spreads on a sphere. In 
