Although the experimental apparatus used by Lagerhjelm for the investigations of 1811–5 resembled that of Polhem a century earlier, there the resemblance ends. Polhem's experimental apparatus seems to show the influence of the French physicist Edmé Mariotte, who
had carried out experimental studies of water and wind mills. Certainly they adhered to the same empirical tradition. Like Mariotte, Polhem was more concerned with articulating and applying generalizations based on experiments than reducing them to fundamental principles; both relied on common sense to guide their reasoning. Although Polhem's work contains an early example of parameter variation and optimization, he was never able to convert his many experimental results into general rules for the design of waterwheels. Nor did he fully appreciate the merit of mathematical analysis of hydrodynamic phenomena. This is evident in his faint praise for the work of his younger colleague Pehr Elvius, who in 1742 published A mathematical treatise on the effect of water mills . Polhem commented in the Proceedings of the Royal Swedish Academy of Sciences: "Although [Elvius'] book is really written for the learned, who are already familiar with the modern mathematics , which by its discoverer the learned Leibniz is called calculus differentialis and by Newton, fluxio curvarum , so does yet Mr. Elvius show his profound knowledge of such puzzling matters, that he gives hope of becoming a good Mechanicus with time, as well in Practice as now to begin with in Theory." What Polhem had considered "modern mathematics" was a standard tool in the hands of the mine official Pehr Lagerhjelm, whose report on the hydrodynamic experiments of 1811–5 financed by the Swedish Ironmasters' Association was a highly mathematical treatise. Lagerhjelm's report also
included a thorough, critical review of relevant international literature. The first fifty pages of the second volume commented on the works of Smeaton, Euler, Borda, Bossut, Banks, Langsdorf, and others.
Lagerhjelm's treatise also evinces a higher level of conceptual awareness. In his preface to the second volume, Lagerhjelm offered an epistemological program to relate theory and experiment for hydrodynamics. His ideas bear a certain resemblance to Kant's theory of knowledge, and the terminology—"phenomenon," "form," and "content"—is similar. For Lagerhjelm, inductive reasoning cannot produce conclusions of universal validity, because "abstractions from a given experience. . .are only valid under the circumstances and within the boundaries essentially associated with the class of phenomena one experienced." The implication was clear: the inductive method followed by Polhem and others, with their thousands of experiments throughout the 18th century, was epistemologically pointless. So, too, was the deductive method, the "speculative root" of knowledge in which Elvius and others had placed their confidence, inadequate in and of itself. The path to truth required a synthesis between "form" and "content," specifically, theory and experiment.