Descartes to Newton

It should here be mentioned that very rude beginnings of the binomial theorem are found very early. The Hindoos and

Arabs used the expansions of $(a+b{)}^2$ and $(a+b{)}^3$ for extracting roots; Vieta knew the expansion of $(a+b{)}^4$; but these

were the results of simple multiplication without the discovery of any law. The binomial coefficients for positive whole exponents were known to some Arabic and European mathematicians. Pascal derived the coefficients from the method of

what is called the "arithmetical triangle." Lucas de Burgo,

Stifel, Stevinus, Briggs, and others, all possessed something from which one would think the binomial theorem could have

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 $\pi$,

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:–- $\pi ={\displaystyle \frac{4}{1+{\displaystyle \frac{1\mathrm{\backslash Fstr}}{2+{\displaystyle \frac{9\mathrm{\backslash Fstr}}{2+{\displaystyle \frac{25\mathrm{\backslash Fstr}}{2+{\displaystyle \frac{49\mathrm{\backslash Fstr}}{2+\text{etc.}\mathrm{\backslash Fstr}}}}}}}}}}}$ 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 logarithmic