Based on electron-positron collision data collected with the BESIII detector operating at the Beijing Electron-Positron Collider II storage rings, the value of R≡σ(e^{+}e^{-}→hadrons)/σ(e^{+}e^{-}→μ^{+}μ^{-}) is measured at 14 center-of-mass energies from 2.2324 to 3.6710 GeV. The resulting uncertainties are less than 3.0% and are dominated by systematic uncertainties.