How to improve Chebyshev bound on the prime counting inequality?

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So, I've understood the proof of A*x/logx < pi(x) < B*x/logx for (A,B) = (0.5,2), but how can I make this difference smaller? Does any one know the methods used by him and further by J.J. Sylvester? I'd like references in english for I found some papers in french, but could figure out anything. Thanks

Ps: pi(x) is the prime counting function