What is the (historical and theoretical) relationship between the scientific experimental method and the two espistemologies of empiricism and rationalism?
I can read here and there that the experimental method is really linked to the philosophy of empiricism. Also, Francis Bacon, the founder of empiricism, is also the founder of the experimental science.
But the physiologist Claude Bernard is also considered the father of the experimental method, and had harsh words against empiricism: "Empiricism is a narrow and abject dungeon from which the imprisoned mind can only escape on the wings of a hypothesis." (Deepl translation checked by myself from French: "L’empirisme est un donjon étroit et abject d’où l’esprit emprisonné ne peut s’échapper que sur les ailes d’une hypothèse").
And we can read elsewhere than Descartes understood the experimental method better than Francis Bacon himself.
Here a quote on the link between empiricism and the scientific method (The Open University: 3 Enlightenment, science and empiricism):
The Enlightenment's dedication to reason and knowledge did not come out of the blue. After all, scholars had for centuries been adding to humanity's stock of knowledge. The new emphasis, however, was on empirical knowledge: that is, knowledge or opinion grounded in experience. This experience might include scientific experiments or firsthand observation or experience of people, behaviour, politics, society or anything else touching the natural and the human. For any proposition to be accepted as true, it must be verifiable, capable of practical demonstration. If it was not so verifiable, then it was an error, a fable, an outright lie or simply a hypothesis. Although Enlightenment thinkers retained a role for theoretical or speculative thought (in mathematics, for example, or in the formulation of scientific hypotheses), they took their lead from seventeenth-century thinkers and scientists, notably Francis Bacon (1561–1626), Sir Isaac Newton and John Locke (1632–1704), in prioritising claims about the truth that were backed by demonstration and evidence. In his ‘Preliminary discourse’ to the Encyclopédie, d'Alembert hailed Bacon, Newton and Locke as the forefathers and guiding spirits of empiricism and the scientific method. To any claim, proposition or theory unsubstantiated by evidence, the automatic Enlightenment response was: ‘Prove it!’ That is, provide the evidence, show that what you allege is true, or otherwise suspend judgement.
And also this quote from a SE user's (@J D) answer (of the question "What does rationalism prove that empiricism can not?"), which also to me points to an association between empiricism and experimental method:
So, a rationalist tends to rely on reason or logic, and an empiricist tends to rely on experimentation or trial and error. Obviously, both are necessary aspects of good thinking if one is interested in connecting one's private world to one's public world. These two traditions in the 21st century have largely developed into an alliance between mathematical and scientific theories across a broad range of disciplines.
Nb: I am not very sure of the difference between "experimental science", "experimental method", and "scientific method".
My question is different from "What does rationalism prove that empiricism can not?" because: (1) It takes here also an historical point of view, and aims to be more precise in terms of which author said what, instead of focusing specifically on the purely philosophical/metaphysical arguments (2) It tackles the terms and concepts of "empirical method", "scientific method", etc. (3) It is not oriented in one way or another
I believe the granularity nature of questions (1) and (2) can be quite beneficial to the community, since I am not sure, as the answers to the question already indicate, that all the facts and terminology at stake are obvious.