Among the many methods devised to compute firing tables, one is to delete or modify certain terms of differential equations of ballistics to make them integrable in finite form. Inaugurated by Johann Bernoulli, this approach was particularly cultivated by Borda, Bézout, Legendre, and Français for the case of air resistance proportional to the square of the velocity, then by other authors for various laws of resistance supposed to correspond better to experience. We study this approach over the 18th and 19th centuries and examine the following questions: what are the numerical tables which were calculated according to this method and to what extent have they been actually used by the artillerymen? How was estimated the mathematical part of the error, coming from the fact that one changes terms in the differential equations? What interactions can be identified between theoretical processes and empirical observations, both in the choice a priori of the simplifications of equations and in the experimental verification of results provided by tables ?