Gregory’s life seems filled with missed opportunities. He invented the first practical reflecting telescope, but it was never built in his lifetime.
An early dispute with Huygens led to a reluctance to publish his work. So it remained overlooked in his lifetime and beyond until in the 1930’s a man called Turnbull discovered his letter and realised just how many original ideas Gregory had developed.
Amongst his mathematical legacy are a number of results that show he was also a first class mathematician. He was a pioneer in the calculation of logarithms, using infinite series for this purpose. Prior to this such calculations involved the extraction of dozens of square roots - and if that ‘extraction’ sounds like teeth being pulled, then it was just as painful. Gregory’s own comment on the power of all previous methods was that, ‘all previous methods have the same ratio to that of the infinite series as the glimmer of dawn has to the splendour of the noonday sun.’.
Born near Aberdeen, he was educated at first by his mother who came from a mathematical family and then at Aberdeen University. He became interested in optics and telescopes and came up with the ingenious idea of using both mirrors and lenses in his design for a telescope.
The advantage in using such a combination is that the optical path can be made to go ‘round corners’ and the telescope can be both more compact and more powerful. It is the basic design of all modern telescopes and binoculars.
He went to London to get a book on the subject published in 1663, and then travelled on to the continent settling in Padua, Italy. Here he published The True Squaring of the Circle and of the Hyperbola.
This book contained significant results in infinite series and was the first systematic book looking at what became calculus. This was also the work which provoked a dispute between Gregory and Huygens.
He returned to Scottish academic life publishing no further works, but crucially he wrote to a friend in London detailing his work.
From these letters we know about his further work on astronomy and mathematical problems which provided solutions to what are known as Taylors expansions (40 years before Taylor) and solved Kepler’s famous circular problem.
He was very close to understanding calculus and discovered infinite series representations for a number of trigonometric functions. He died at the age of just 37 and it was to be 250 years later that his optical work was truly appreciated.
To calculate the logarithm of 2 (in the exponential base), he used,
and he quickly found the approximate answer,
Using the same technique, he was led to this amazing identity for \pi:
The infinite series that Gregory discovered were the later cornerstones of Calculus.