James Gergory

James Gregory was born in the Manse of Drumoak. This is a small parish on the river Dee, about fifteen kilometres west of Aberdeen. His father was John Gregory and his mother was Janet Anderson. John Gregory had studied at Marischal College in Aberdeen, then gone on to study theology at St Mary's College in the University of St Andrews before spending his life in the parish of Drumoak. Turnbull writes [20]:-

[John Gregory] was a man of courage and foresight but was not conspicuous for outstanding intellectual gifts ...

James seems to have inherited his genius through his mother's side of the family. Janet Anderson's brother, Alexander Anderson, was a pupil of ViЁЁte. He acted as an editor for ViЁЁte and fully incorporated ViЁЁte's ideas into his own teaching in Paris. James was the youngest of his parents three children. He had two older brothers Alexander (the eldest) and David, and there was an age gap of ten years between James and David.

James learnt mathematics first from his mother who taught him geometry. His father John Gregory died in 1651 when James was thirteen and at this stage James's education was taken over by his brother David who was about 23 at the time. James was given Euclid's Elements to study and he found this quite an easy task. He attended Grammar School and then proceeded to university, studying at Marischal College in Aberdeen.

Gregory's health was poor in his youth. He suffered for about eighteen months from the quartan fever which is a fever which recurs at approximately 72-hour intervals. Once he had shaken off this problem his health was good, however, and he wrote some years later that the quartan fever (see for example [20]):-

... is a disease I am happily acquainted with, for since that time I never had the least indisposition; nevertheless that I was of a tender and sickly constitution formerly.

Gregory began to study optics and the construction of telescopes. Encouraged by his brother David, he wrote a book on the topic Optica Promota. In the preface he writes:-

Moved by a certain youthful ardour and emboldened by the invention of the elliptic inequality, I have girded myself with these optical speculations, chief among which is the demonstration of the telescope.

The reader may not understand Gregory's reference to "the elliptic inequality" which in fact refers to Kepler's discoveries. Gregory, in Optica Promota, describes the first practical reflecting telescope now called the Gregorian telescope.

The book begins with 5 postulates and 37 definitions. He then gives 59 theorems on reflection and refraction of light. There follows propositions on mathematical astronomy discussing parallax, transits and elliptical orbits. Next Gregory gives details of his invention of a reflecting telescope. A primary concave parabolic mirror converges the light to one focus of a concave ellipsoidal mirror. Reflection of light rays from its surface converge to the ellipsoid's second focus which is behind the main mirror. There is a central hole in the main mirror through which the light passes and is brought to a focus by an eyepiece lens. The tube of the Gregorian telescope is thus shorter than the sum of the focal lengths of the two mirrors. His novel idea was to use both mirrors and lenses in his telescope. He showed that the combination would work more effectively than a telescope which used only mirrors or used only lenses.

The book was only a theoretical description of the telescope for at this stage one had not been constructed. Gregory remarks in the book [21]:-

... on his lack of skill in the technique of lens and mirror making ...

In 1663 Gregory went to London. There he met Collins and a lifelong friendship began. One of Gregory's aims was to have Optica Promota published and he achieved this. His other aim was to find someone who could construct a telescope to the design set out in his book. Collins advised him to seek the help of a leading optician by the name of Reive who, at Gregory's request, tried to construct a parabolic mirror. His attempt did not satisfy Gregory who decided to give up the idea of having Reive construct the instrument. However, Hooke learnt of Reive's failed attempt at making the parabolic mirror and this would lead to a successful construction of the first Gregorian telescope around ten years later.

In London Gregory also met Robert Moray, president of the Royal Society, and Moray attempted to arrange a meeting between Gregory and Huygens in Paris. However, Huygens was not in Paris and the meeting did not materialise. Moray was to play a major role in Gregory's career somewhat later.

In 1664 Gregory went to Italy. He visited Flanders, Rome and Paris on his journey but spent most time at the University of Padua where he worked on using infinite convergent series to find the areas of the circle and hyperbola. At Padua he worked closely with Angeli whose [20]:-

... teaching profoundly influenced Gregory, particularly in providing the twin keys to the calculus, the method of tangents (differentiation) and of quadratures (integration).

In Padua Gregory was able to live in the house of the Professor of Philosophy who was Professor Caddenhead, a fellow Scot. Two works which were published by Gregory while he was in Padua are Vera circuli et hyperbolae quadratura published in 1667 and Geometriae pars universalis published right at the end of his Italian visit in 1668.

Of Vera circuli et hyperbolae quadratura Dehn and Hellinger write in [5]:-

In this work Gregory lays down exact foundations for the infinitesimal geometry then coming into existence. It is remarkable that some decades later, at the time when analysis was in a state of revolutionary development, exactness was at a much lower standard than with Gregory, and generally with the authors writing before the discoveries of Newton and Leibniz (e.g. Huygens, Mengoli, Barrow).

The work we are dealing with is of quite a different character. On the one hand, the source from which he is getting his inspiration is quite unknown to us. On the other hand we find here a singular mixture of far-reaching ideas, exact methods, incomplete deductions, and even false conclusions.

The work was really trying to prove that ¦Ð and e are transcendental but Gregory's arguments contain a subtle error. However, this should not in any way detract from the brilliance of the work and the amazing collection of ideas which