Sixty-four point angular distributions for the elastic scattering of α particles by 40Ca have been measured from 12 to 18 MeV in 100-keV intervals. The angular range is from 26.7° to 176.1° (c.m.). Optical-model fits to these data have been obtained, using a Woods-Saxon real potential and a surface imaginary potential with a form given by the derivative of a Woods-Saxon function. The real well-radius value of 5.2 F has been taken from previous analyses of electron scattering data. A consistent set of the other geometric parameters was chosen by fitting a representative set of five angular distributions covering the entire energy range. With the geometric parameters thus fixed, two-parameter fits to all of the data have been obtained in which the real and the imaginary potential strengths are varied. A fitting criterion which compensates for the large Rutherford contribution to the scattering at forward angles is introduced to improve the fits. Of the ten phase-equivalent potentials with real potential strengths less than 210 MeV, the three with average values of 25, 105, and 132 MeV are traced as functions of energy. A cluster-model calculation which places a bound α-particle state at -5.23 MeV in 44Ti yields a potential-well depth of 136 MeV, in rather good agreement with the 132-MeV average value found in the scattering analysis.