GEORGE BRUCE HALSTED — NEW ELEMENTS OF GEOMETRY. 
11 
the ordinary g-eometry we find 4rcr 2 as magnitude of a sphere of 
radius r, whence the force must diminish in the squared ratio of 
the distance. 
In the imaginary g-eometry I have found the surface of the 
sphere equal to 
7T (e r —e~ r ) 2 , 
and possibly the molecular forces may follow such a g-eometry, 
whose whole diversity would depend, consequently, on the num- 
ber, always very great, e. 
Moreover, suppose this only a mere hypothesis, for whose con- 
firmation other more convincing- proofs are to seek; yet neverthe- 
less it can not be doubted that forces alone produce all: motion, 
velocity, time, mass, even distances and angles. 
With forces all are in a close connection, which we do not under- 
stand in its essence, wherefore also we can not afhrm, that in the 
relation of different kinds of magriitudes to one another only their 
ratios can enter. 
If we admit the dependence on the ratio, why should we not also 
assume a direct dependence? 
Certain circumstances already favor this opinion. For example, 
the magnitude of attractive force has as expression the mass 
divided by the square of the distance. For the distance null this 
expression really represents nothing-. 
We must begin with some, larg-e or small, but always really 
present distance, and then first the force manifests itself. Now 
we may ask how distance produces this force? How a bond be- 
tween thing's so essentially different can exist in nature? 
That we will probably never understand. But if it is true that 
forces depend upon distances, just so also may lines be dependent 
on angles. At least the diversity is alike in the two cases, for the 
difference lies properly not in the idea, but only in this, that we 
know the one dependence from experience, but the other, for want 
of researches, we must mentally assume, either beyond the limits 
of the visible world, or in the narrow sphere of molecular attrac- 
tions. 
However that may be, yet will the assumption, that merely the 
ratio of the distances can determine the angles, be a special case, 
at which we always arrive, if we assume the lines infinitesimal. 
The procedure of ordinary g-eometry therefore always leads to 
results true but not in so wide a sense as those given by the gen- 
eral g-eo metric system which I have called imaginary g-eometry. 
The difference between the equations of the one and the other 
comes from the introduction of a new constant, which observations 
