Calculus of Finite Differences
in the Long Nineteenth Century.
Most material in Net Advance Retro
antedates 1920. It may be obsolete or incorrect.
 General:

Recherches sur l'intégration des équations différentielles
aux différences finies et sur leur usage dans la théorie des hasards
by Pierre Simon Laplace
[Mémoires de l'Académie Royale des Sciences de Paris,
(Savants étrangers) , (1773)]

On the developement of exponential functions; together with
several new theorems relating to finite differences
by J. F. W. Herschel
[Phil. Trans. R. S. L. 106, 25 (1816)]

On circulating functions, and on the integration of a class
of equations of finite differences into which they enter
as coefficients
by J. F. W. Herschel
[Phil. Trans. R. S. L. 108, 144 (1818)]

A Collection of Examples of the Applications of the
Calculus of Finite Differences
by J. F. W. Herschel
[Cambridge: Deighton, 1820]

An Essay on Algebraic Development ... a New and Comprehensive Notation
by Rev. Thos. Jarrett
[Cambridge: Deightons, 1831]

A Treatise on Differential Equations, and on the Calculus
of Finite Differences
by J. Hymers
[Cambridge: Deighton and Deighton, 1839]

General Methods in Analysis for the resolution of Linear
Equations in Finite Differences and Linear Differential
Equations
by Chas. Jas. Hargreave
[Phil. Trans. R. S. L. 140, 261 (1849)]

Calculus of Finite Differences
by George Boole
[Cambridge: Macmillan, 1860]

On Some Problems in the Calculus of Finite Differences
by Wm. Rae Macdonald
[Edinburgh: Actuarial Society, 1876]

Notes on Finite Differences. For the Use of Students of the
Institute of Actuaries
by A. W. Sunderland
[London: C. & E. Layton, 1885]

Elements of Finite Differences, also Solutions to Questions
Set for Part I of the Examinations of the Institute of
Actuaries
by J. Burn and E. H. Brown
[London: C. & E. Layton, 1902]
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