**The catenary is formed as the arithmetic mean between two curves which Leibniz called ``logarithmic,'' and are today called exponential. In the figure, the blue lines are spaced equally along a horizontal axis. The ``logarithmic'' curve is formed by the vertical lengths which are in geometric proportion. If OO=x**^{0 }=1, then ; e'=x^{1} and e= x^{-1}=1/x^{1}; d'=x^{2} and d=x^{-2}=1/x^{2}, etc. The catenary is formed by adding length e to e' and dividing the combined length by two; then adding length d to d' and dividing the combined length by two, etc. The points of the catenary are equal to (x^{n}+1/x^{n})/2.