Unsubtracted dispersion relations are used to link new data obtained in the time-like region with the electromagnetic structure of the nucleons as measured in the space-like region. Simple analytic representations of the magnetic form factor of the proton GpM( Q2) and the electric form factor of the neutron GnE( Q2) are derived which have the following properties built-in: the correct normalization and slope at Q2 = 0 and correct asymptotic behavior at | Q2| → ∞. The pred of perturbative QCD at asymptotic Q2 and meson physics at low Q2 are combined to fit the existing data in the entire range of the four-momentum transfer squared. This leads to the following observations: (i) Measurements of cross sections for the reaction p overlinep → e +e - taken at the Fermilab E-760 match measurements made at the SLAC-136 experiment only if the logarithmic falloff of Q4GpM( Q2), expected from leading twist predictions of perturbative quantum chromodynamics, is included. (ii) Analytic continuation of the neutron time-like form factor into the space-like interval 0.1 GeV 2 ≤ Q2 ≤ 1 GeV 2 using the limited FENICE data gives a factor of ∼ 2 inconsistency in GnE( Q2) with the recent Saclay and Mainz measurements.