The discovery of the important neutron-rich nucleus _{20}^{60}Ca_{40} and seven others near the limits of nuclear stability is reported from the fragmentation of a 345 MeV/u ^{70}Zn projectile beam on ^{9}Be targets at the radioactive ion-beam factory of the RIKEN Nishina Center. The produced fragments were analyzed and unambiguously identified using the BigRIPS two-stage in-flight separator. The eight new neutron-rich nuclei discovered, ^{47}P, ^{49}S, ^{52}Cl, ^{54}Ar, ^{57}K, ^{59,60}Ca, and ^{62}Sc, are the most neutron-rich isotopes of the respective elements. In addition, one event consistent with ^{59}K was registered. The results are compared with the drip lines predicted by a variety of mass models and it is found that the models in best agreement with the observed limits of existence in the explored region tend to predict the even-mass Ca isotopes to be bound out to at least ^{70}Ca.