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. OO=1; e'=OO2 and e=1/OO2; d'=OO3 and d=1/OO3, 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 (OOn+1/OOn)/2.