The elements of the geometry of spherical space are most easily
obtained by putting for space of four dimensions the equation for the
sphere
_x_² + _y_² +_z_² + _t_² = _R_² (1.)
and for the distance _ds_ between the points (_x_, _y_, _z_, _t_) and
[(_x_ + _dx_) (_y_ + _dy_) (_z_ + _dz_) (_t_ + _dt_)] the value
_ds_² = _dx_² + _dy_² + _dz_² + _dt_² (2.)
It is easily found by means of the methods used for three dimensions
that the shortest lines are given by equations of the form
_ax_ + _by_ + _cz_ + _ft_ = 0 }
} (3.)
_αx_ + _βy_ + _γz_ + _φt_ = 0 }
in which _a_, _b_, _c_, _f_, as well as _α_, _β_, _γ_, _φ_, are
constants.