The modified Abel lemma on summation by parts with the ``remainder term'' is employed to investigate the partial sums of a quadratic $_7F_7$-series. Several unusual transformation formulae for these sums are established. As consequences, some new transformations of quadratic $_7F_6$-series are deduced, especially two of which respectively generalize two known $_3F_2(1)$-series summation formulae due to Watson and Whipple (1925).