  We obtain extensions of classical hypergeometric identities of Bailey and
Whipple that transform nearly-poised and very-well-poised series to
Saalsch\"utzian series, Saalsch\"utzian series to Saalsch\"utzian series, and
very-well-poised and nearly-poised series to very-well-poised series. We employ
a method in which summations and transformations of lower-order series are used
to obtain transformations of higher-order series. By taking limits, we also
obtain extensions of two classical quadratic transformations of Whipple and
Bailey. Furthermore, we show how a number of other well-known results regarding
hypergeometric series follow as special cases of our results.
