Throughout the 18th and 19th centuries, various methods were used to integrate the ballistic differential equation and to calculate firing tables for the needs of artillery. Mathematicians, military engineers and artillerymen, although belonging to different milieus and different countries, collaborated and inspired themselves mutually through a regular interaction between analytic theory of differential equations, numerical and graphical techniques of integration, and empirical research by means of experimental measures. This research on the ballistic equation played the role of a laboratory where scientists have been able to test all possible approaches for calculating the solution of a differential equation. We will show how these trials, on the one hand, have allowed to improve some important points in the abstract theory of differential equations and, on the other hand, have contributed to give birth to numerical analysis as an autonomous discipline at the beginning of the 20th century. On the social and institutional point of view, the ballistic problem will permit us to analyze the links between civil and military scientific research in Europe at the time, and to put in evidence some differences in the national approaches observed in France, Germany, England and Italy.