This monograph explores the history of the contribution to ballistics by the American mathematician Gilbert Ames Bliss during World War I.  Drawing on the then-evolving calculus of variations, Bliss pioneered a novel technique for solving the problem of differential variations in ballistic trajectory.  Called Bliss’ adjoint method, this technique was both hailed and criticized at the time: it was seen as both a triumphant application of pure mathematics to an applied problem and as a complex intrusion of higher mathematics into the jobs of military personnel not particularly interested in these matters.  Although he received much praise immediately after the War, the details of Bliss’ work, its furthering of pure mathematical thought, and its absorption into mainstream ballistic work and instruction have never been adequately examined. Gluchoff explores the mathematics of Bliss’ work and the strands from which his technique was developed.  He then documents the efforts to make the adjoint method accessible to military officers and the conflicts that emerged as a result both between mathematicians and officers and among mathematicians themselves.  The eventual absorption of the adjoint method into range firing table construction is considered by looking at later technical books which incorporate it, and, finally, its influence on the ongoing development of functional calculus is detailed. From Frechet Differentials to Firing Tables will appeal to historians of mathematics, physics, engineering, and warfare, as well as current researchers, professors, and students in these areas.