- Q: What is meaning of the bold clause? Is it a part of what Peirce labeled as an a priori method?
Pierce labeled a combination of two things. One of these two things is the use of reason. The other is a developing consensus among those debating the merits of one belief over another. This use and this consensus combine to form the thing that Pierce labeled the a priori method for acquiring knowledge.
- Q: Is "developing" an adjective? Does it mean growing or improving? Can I eliminate it in translation?
A: Maybe, yes, and no.
Just like "growing" and "improving", "developing" is a participle. I don't call it an adjective in this use, but some people do. It is certainly doing the same job that an adjective typically does. Here, it modifies "consensus".
"Developing" implies more than "growing and improving". It implies that someone is actively trying to make improvements and extensions. In this context, the someone is clearly represented: People who debate the merits of one belief over another develop a consensus.
The nouns "use" and "consensus" are grammatically parallel. They are, after all, both abstract nouns. However, we can paraphrase the author's sentiment while establishing a closer semantic parallel:
Peirce labeled the use of reason, and the development of consensus among those debating the merits of one belief over another, the a priori method for acquiring knowledge.
Here, "use" and "development" have even more in common than "use" and "consensus". Both "use" and "development" represent methods, or abstract actions. Combined, they still represent a method. This is the sort of thing that makes sense as "the a priori method for acquiring knowledge".
The semantics carried by "developing" are essential to the author's intended meaning. You'll want to avoid letting that get lost in translation.