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The Baconian method is the investigative method developed by Sir Francis Bacon. The method was put forward in Bacon's book Novum Organum (1620), or 'New Method', and was supposed to replace the methods put forward in Aristotle's Organon. This method was influential upon the development of the scientific method in modern science; but also more generally in the early modern rejection of medieval Aristotelianism. With the upcoming Romanticism in the 19th century, it was replaced by Humboldtian science. 
Bacon's method is an example of the application of inductive reasoning. By reasoning using "induction", Bacon meant the ability to generalize a finding stepwise, based on accumulating data. He advised proceeding by this method, or in other words, by building a case from the ground up. He wrote in the Novum Organum that
"Our only hope, then is in genuine Induction... There is the same degree of licentiousness and error in forming Axioms, as in abstracting Notions: and that in the first principles, which depend in common induction. Still more is this the case in Axioms and inferior propositions derived from Syllogisms." (See for example, aphorism XVII of the Novum Organum.)
The method consists of procedures for isolating and further investigating the form nature, or cause, of a phenomenon, including the method of agreement, method of difference, and method of concomitant variation.
Bacon suggests that you draw up a list of all things in which the phenomenon you are trying to explain occurs, as well as a list of things in which it does not occur. Then you rank your lists according to the degree in which the phenomenon occurs in each one. Then you should be able to deduce what factors match the occurrence of the phenomenon in one list and don't occur in the other list, and also what factors change in accordance with the way the data had been ranked.
Thus, if an army is successful when commanded by Essex, and not successful when not commanded by Essex: and when it is more or less successful according to the degree of involvement of Essex as its commander, then it is scientifically reasonable to say that being commanded by Essex is causally related to the army's success.
From this Bacon suggests that the underlying cause of the phenomenon, what he calls the "form," can be approximated by interpreting the results of one's observations. This approximation Bacon calls the "First Vintage." It is not a final conclusion about the formal cause of the phenomenon but merely a hypothesis. It is only the first stage in the attempt to find the form and it must be scrutinized and compared to other hypotheses. In this manner, the truth of natural philosophy is approached "by gradual degrees," as stated in his Novum Organum.
The "Baconian method" does not end at the First Vintage. Bacon described numerous classes of Instances with Special Powers, cases in which the phenomenon one is attempting to explain is particularly relevant. These instances, of which Bacon describes 27 in the Novum Organum, aid and accelerate the process of induction.
Aside from the First Vintage and the Instances with Special Powers, Bacon enumerates additional "aids to the intellect" which presumably are the next steps in his method. These additional aids, however, were never explained beyond their initial limited appearance in Novum Organum.
The Natural History of Pliny the Elder was a classical Roman encyclopedia work. Induction, for Bacon's followers, meant a type of rigour applied to factual matters. Reasoning should not be applied in plain fashion to just any collection of examples, an approach identified as "Plinian". In considering natural facts, a fuller survey was required to form a basis for going further. Bacon made it clear he was looking for more than "a botany" with discursive accretions.
In concrete terms, the cabinet of curiosities, exemplifying the Plinian approach, was to be upgraded from a source of wonderment to a challenge to science. The main source in Bacon's works for the approach was his Sylva Sylvarum, and it suggested a more systematic collection of data in the search for causal explanations.
Underlying the method, as applied in this context, are therefore the "tables of natural history" and the ways in which they are to be constructed. Bacon's background in the common law has been proposed as a source for this concept of investigation.
As a general intellectual programme, Bacon's ideas on "natural history" have been seen as a broad influence on British writers later in the 17th century, in particular in economic thought and within the Royal Society.
Bacon also listed what he called the idols (false images) of the mind. He described these as things which obstructed the path of correct scientific reasoning.
Thomas Browne the physician (1605–82) was one of the first writers scientists to adhere to the empiricism of the Baconian method. His encyclopaedia Pseudodoxia Epidemica (1646–76) includes numerous examples of Baconian investigative methodology; and its preface paraphrases lines from Bacon's essay On Truth in his 1605 work The Advancement of Learning. Isaac Newton's saying hypotheses non fingo (I don't frame hypotheses) was occurs in later editions of the Principia. It represents his preference for rules that could be demonstrated, as opposed to unevidenced hypotheses.
The Baconian method was further developed and promoted by John Stuart Mill. His 1843 book, A System of Logic, was an effort to shed further light on issues of causation. In this work, he formulated the five principles of inductive reasoning now known as Mill's methods.