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Here is a video explaining some of Auguste Comte's theories about Positive Philosophy. https://www.youtube.com/watch?v=9yCpTe9e-MI
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Started working for Henri De Saint-Simon as his secretary which put him into the political spot light. He was able to publish many articles at this time, which he later re-published. Henri De Saint-Simon chose to work with Comte due to his view of the social construct of society. You could say that this was the start to his career. (Stanford.edu)
https://plato.stanford.edu/entries/comte/#ForYeaColSaiSimEarWri -
He began teaching the Course of Positive Philosophy. He taught this course for many years. This is his most famous piece of work. He published many works during this time. He had two goals with this course, one was setting the foundation for sociology, and "coordination of the whole of positive knowledge." (standford.edu)
https://plato.stanford.edu/entries/comte/#ForYeaColSaiSimEarWri -
Comte published is only Mathematical work at this time. "Elementary Trieste on Analytic Geometry" It would seem that he did this not because he enjoyed math, but simply because math has a set of rules and it has its true answers. (Stanford.edu)
https://plato.stanford.edu/entries/comte/#ForYeaColSaiSimEarWri -
Published "Philosophical Treatise on Popular Astronomy"
He also published "Discourse on the Positive Spirit" He used as the preface to the treatise on Astronomy this "marked the change of direction by its emphasis on the moral dimension of the new philosophy." (Standford.edu)
https://plato.stanford.edu/entries/comte/#ForYeaColSaiSimEarWri -
Auguste Comte died in 1857 and "He is buried in the Père-Lachaise cemetery" (stanford.edu)
https://plato.stanford.edu/entries/comte/#ForYeaColSaiSimEarWri