The story of the rise and development of multistep methods to integrate ordinary differential equations has been somewhat neglected so far.These methods may be viewed as originally stemming from the Gregory-Newton formula, involving quadratures, this being applied first by Clairaut to a system of differential equations, in his calculations of the motion of Halley's comet on its return in 1759. This pointed the way for the rise of multistep methods proper, these subsequently being arrived at independently at various times, by J.C. Adams (1855), G.H. Darwin (1897), W.F. Sheppard (1899), and C. Størmer (1907). These methods called for the computation of massive numerical tables, to cater for the requirements of highly involved problems in applied mathematics. These methods, originating as they did in the mathematical skills of British astronomers, stand out as the apogee of one era in the history of numerical analysis.