been gotten with a little attention, "if we did not know that such simple relations were difficult to discover."
Though Wallis had obtained an entirely new expression for , he was not satisfied with it; for instead of a finite number of terms yielding an absolute value, it contained merely an infinite number, approaching nearer and nearer to that value. He therefore induced his friend, Lord Brouncker (1620?-1684), the first president of the Royal Society, to investigate this subject. Of course Lord Brouncker did not find what they were after, but he obtained the following beautiful equality:
Continued fractions, both ascending and descending, appear to have been known already to the Greeks and Hindoos, though not in our present notation. Brouncker's expression gave birth to the theory of continued fractions.
Wallis' method of quadratures was diligently studied by his disciples. Lord Brouncker obtained the first infinite series for the area of an equilateral hyperbola between its asymptotes. Nicolaus Mercator of Holstein, who had settled in England, gave, in his Logarithmotechnia (London, 1668), a similar series. He started with the grand property of the equilateral hyperbola, discovered in 1647 by Gregory St, Vincent, which connected the hyperbolic space between the asymptotes with the natural logarithms and led to these logarithms being called hyperbolic. By it Mercator arrived at the logarithmic series, which Wallis had attempted but failed to obtain. He showed how the construction of logarith-
