James Gregory and MacLaurin series

James Gregory was a Scottish mathematician and astronomer I never heard about until recently. Born in 1638 his life turned out to be short. At the age of 36 he suffered a stroke while watching the skies. He died a few years later. That must be the shortest biography ever.

In his time he maintained friendly connections with Isaac Newton. Like Newton, he was a Fellow of the Royal Society. He was also the uncle of a younger James Gregory. His nephew is know for tutoring Colin MacLaurin, the Scottish mathematician know for the MacLaurin Series.

But in fact it was James Gregory who first studied these series. They are a special case of the Taylor Series. Brook Taylor, from Middlesex, was born in 1685, well after Gregory’s death. The Taylor Power Series can describe functions that can be derived infinitely. Examples are e^x or the trigonometric functions. The Taylor Series can be written as

    \[f(x)=\sum_{n=0}^\infty\frac{f^{(n)}a}{n!}(x-a)^n\]

The principle is to first estimate the function by starting with a constant function f(x)=a. Then the Taylor Series starts integrating each part again and again to a higher power. In this way the resulting power series approximates the original function better and better; first around a, in infinity everywhere.

The MacLaurin Series is a special case of the Taylor Series. It develops not around a, but around 0, and so becomes

    \[f(x)=\sum_{n=0}^\infty\frac{f^{(n)}0}{n!}(x)^n\]

A very handy tool to evaluate functions like e^x, \sin and \cos. But it can also come in handy integrating functions that have no solution closed form, like \int \cos(x^2). There’s no way to integrate it directly: you’ll always end up with a new integral of the derivative as part of the answer.

But if we look at the solution with the help of the Taylor Series we get for \cos(x)

    \[\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+....\]

In sigma notation this is

    \[\cos(x)=\sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}\]

For \cos(x^2) we get after substitution

    \[\cos(x^2)=\sum_{n=0}^\infty (-1)^n\frac{x^{4n}}{(2n)!}\]

Now we can integrate:

    \[\int\sum_{n=0}^\infty (-1)^n\frac{x^{4n}}{(2n)!}\mathrm{d}x=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}\int x^{4n}\mathrm{d}x=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}\frac{x^{4n+1}}{4n+1}+C\]

This kind of integral is also known as Fresnelintegral. Two Fresnelintegrals are

    \[C(x)=\int_0^x\cos(t^2)\]

en

    \[S(x)=\int_0^x\sin(t^2)\]

These are important integrals in optics. They have to be integrated at the interval \{0,x\} as you can see.

Back to James Gregory. Unknowing of its existence, Colin MacLaurin reproduced the work of Gregory, who developed this series much earlier. He described the principle in his work Vera Circuli et Hyperbolae Quadratura from 1667.

No doubt his nephew must have known the principles while tutoring MacLaurin.

James Gregory wasn’t just a mathematician, he was an astronomer as well. He developed his own telescope design, the Gregorian reflector. It is similar in design to the Cassegrain telescope. While that telescope has a convex secondary mirror, sending the light through a hole in the center of the primary mirror, the Gregorian telescope has a concave mirror. Modern Schmidt-Cassegrain type telescope also have a corrector plate. Optics are never easy, as the development of telescopes show.

It might be cynical that he died while watching the Moon. But then again, he might have lived longer and died while doing mathematics. Dying way too early, at 36, all I can say is that it’s the travel that counts. Not to mention the results.