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The missing fractions in Brouncker's sequence of continued fractions for π

The forgotten continued fractions Lord Brouncker's continued fraction for π is (In this paper we will use the more convenient notation This fraction first appeared in the Arithmetica Infinitorum [1] by John Wallis. In this book, along with topics that lead to Newton's calculus, Wallis derives his famous product Wallis tells us that he showed this product to Lord Brouncker, and in turn Brouncker converted it not only into (1), but into an infinite sequence of continued fractions.