We perform a fit to measurements of the cosmic ray spectrum in the energy range between 10 15 eV and 10 18 eV using data from the TALE, Tunka and Auger experiments. We also fit the data on the depth of shower maximum, Xmax, from Tunka and Yakutsk or from Auger to constrain the cosmic ray composition. We consider a Galactic component that is a mixture of five representative nuclear species (H, He, N, Si and Fe), for which we adopt rigidity dependent broken power-law spectra, and we allow for an extragalactic component which becomes strongly suppressed for decreasing energies. The relative abundances of the Galactic components at 10 15 eV are taken to be comparable to those determined by direct measurements at 10 13 eV. The main features of the spectrum and of the composition are reproduced in these scenarios. The spectral knee results from the break of the H spectrum at E k ≃ 3 × 10 15 eV, although it is broadened by the comparable contribution from He which has a break at about 6×10 15 eV. The low-energy ankle at E la ≃ 2×10 16 eV coincides with the strong suppression of the H and He Galactic components and the increasing relative contribution of the heavier ones, but the observed hardening of the spectrum at this energy turns out to result from the growing contribution of the extragalactic component. The second-knee at E sk ≃ 26E k ≃ 8 × 10 16 eV is associated with the steepening of the Galactic Fe component. The transition to the regime in which the total cosmic ray flux is dominated by the extragalactic component takes place at an energy of about 10 17 eV. The parameters of the fit depend on the specific Xmax dataset that is considered, with the Yakutsk and Tunka data leading to a suppression of the light components being steeper beyond the knee so as to allow a faster growth of the average mass around the low-energy ankle. The results also depend on the hadronic model that is used to interpret Xmax measurements and we compare the parameters obtained with Sibyll 2.3, EPOS-LHC and QGSJET II-04. The impact of the possible existence of a maximum rigidity cutoff in the Galactic components is also discussed.