**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]:-

[ames seems to have inherited his genius through his mother's side of the family. Janet Anderson's brother, Alexander Anderson, was a pupil of Viète. He acted as an editor for Viète and fully incorporated Viète'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.John Gregory]was a man of courage and foresight but was not conspicuous for outstanding intellectual gifts ...

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]):-

Gregory began to study optics and the construction of telescopes. Encouraged by his brother David, he wrote a book on the topic... 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.

*Optica Promota*. In the preface he writes:-

The reader may not understand Gregory's reference to "the elliptic inequality" which in fact refers to Kepler's discoveries. Gregory, inMoved 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.

*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]:-

In 1663 Gregory went to London. There he met Collins and a lifelong friendship began. One of Gregory's aims was to have... on his lack of skill in the technique of lens and mirror making ...

*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]:-

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

*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]:-

The work was really trying to prove that π 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 it contains such as: convergence, functionality, algebraic functions, transcendental functions, iterations etc.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.

Before he left Padua Gregory published *Geometriae pars universalis* which is really [13]:-

This book contained the first known proof that the method of tangents (differentiation in our modern terminology) was inverse to the method of quadratures (integration in our modern terminology). Gregory shows how to transform an integral by a change of variable and introduces the... the first attempt to write a systematic text-book on what we should call the calculus.

*x*↦

*x*- 0(

*x*) idea which is the basis of Newton's fluxions. Perhaps it is worth saying a little about how Gregory's work relates to that of Newton. By the time that Gregory published this work Newton had formed his ideas of the calculus so probably had not been influenced by Gregory. On the other hand Newton had not said anything of his ideas and so certainly these ideas could not have influenced Gregory. Essentially Newton and Gregory were working out the basic ideas of the calculus at the same time, as, of course, were other mathematicians.

Gregory returned to London from Italy at about Easter 1668. He had sent a copy of *Vera circuli et hyperbolae quadratura* to Huygens and written a covering letter saying how he was looking forward to hearing the expert opinions of Huygens on it. Huygens did not reply but published a review of the work in July 1668. In the review he raised some objections and also claimed that he had been the first to prove some of the results. On the one hand the summer months that Gregory spent in London were profitable, particularly through his friendship with Collins. It was a time of rapid mathematical development and Gregory found that Collins, with his up-to-date knowledge of developments, was most helpful to him. On the other hand he was upset by Huygens' comments which he took to imply that Huygens was accusing him of stealing his results without acknowledgement.

It was indeed unfortunate that these two great mathematicians should enter into a dispute, although having said that it is worth noting that disputes were common at this time, particularly regarding priority. Looking at the dispute with the hindsight of today's understanding of the mathematics involved we can say that Huygens was certainly unfair in suggesting that Gregory had stolen his results. Gregory had proved them independently and Huygens should have realised that Gregory could not have known of them. However, Huygens' main mathematical objection to Gregory's proof is a valid one. Despite there is brilliant work in this text and in [16] Scriba shows how close Gregory was to making further major discoveries. He writes [16]:-

The dispute had another unfortunate consequence, namely that Gregory became much less keen to announce the methods by which he made his mathematical discoveries and, as a consequence, it was not until Turnbull examined Gregory's papers in the library in St Andrews in the 1930s that the full brilliance of Gregory's discoveries became known.Clearly[Gregory]could not see the consequences that lay concealed in his construction. But he had an unerring sense of where they would lead ...

We can now be certain that during the summer of 1668 Gregory was completely familiar with the series expansions of sin, cos and tan. He also established that

*x*

*dx*= log(sec

*x*+ tan

*x*)

*Exercitationes Geometricae*as a counterattack on Huygens. Although he did not disclose his methods in the small treatise he discussed topics including various series expansions, the integral of the logarithmic function, and other related ideas.

Also during his time in London in the summer of 1668 Gregory attended meetings of the Royal Society and he was elected a fellow of the Society on 11 June of that year. He presented various papers to the Society on a variety of topics including astronomy, gravitation and mechanics. We have already mentioned that Robert Moray was a member of the Royal Society with whom Gregory was friendly. Moray was a fellow Scot and a graduate of St Andrews. It is almost certain that it was through Moray that Charles II was persuaded to create the Regius Chair of Mathematics in St Andrews, principally to allow Gregory a position in which he could continue his outstanding mathematical research.

Gregory arrived in St Andrews late in 1668. He was not attached to a College, as were the other professors, but given the Upper Hall of the university library as his place of work. It was the only university building which was not part of a college so was the only possible place for an unattached professor. Gregory found that St Andrews was of classical outlook where the latest mathematical work was totally unknown. In 1669, not long after arriving in St Andrews, Gregory married Mary Jamesone who was a widow. They had two daughters and one son. After Gregory's death she married for a third time.

While in St Andrews Gregory gave two public lecture each week which were not well received:-

However Gregory was to carry out much important mathematical and astronomical work during his six years in the Regius chair. He kept in touch with current research by corresponding with Collins. Gregory preserved all Collins' letters, writing notes of his own on the backs of Collins' letters. These are still preserved in the St Andrews University library and provide a vivid record of how one of the foremost mathematicians of his day made his discoveries.... I am often troubled with great impertinences: all persons here being ignorant of these things to admiration.

Collins sent Barrow's book to Gregory and, within a month of receiving it, Gregory was extending the ideas in it and sending Collins results of major importance. In February 1671 he discovered Taylor series (not published by Taylor until 1715), and the theorem is contained in a letter sent to Collins on 15 February 1671. The notes Gregory made in discovering this result still exist written on the back of a letter sent to Gregory on 30 January 1671 by an Edinburgh bookseller. Collins wrote back to say that Newton had found a similar result and Gregory decided to wait until Newton had published before he went into print. He still felt badly about his dispute with Huygens and he certainly did not wish to become embroiled in a similar dispute with Newton.

The feather of a sea bird was to allow Gregory to make another fundamentally important scientific discovery while he worked in St Andrews. The feather became the first diffraction grating but again Gregory's respect for Newton prevented him going further with this work. He wrote:-

The Upper Room of the library had an unbroken view to the south and was an excellent site for Gregory to set up his telescope. Gregory hung his pendulum clock on the wall beside the same window. The clock, made by Joseph Knibb of London, was purchased in 1673. Huygens patented the idea of a pendulum clock in 1656 and his work describing the theory of the pendulum was published in 1673, the year Gregory purchased his clock.Let in the sun's rays by a small hole to a darkened house, and at the hole place a feather(the more delicate and white the better for this purpose), and it shall direct to a white wall or paper opposite to it a number of small circles and ovals(if I mistake them not)whereof one is somewhat white(to wit, the middle which is opposite the sun)and all the rest severally coloured. I would gladly hear Mr Newton's thoughts of it.

In 1674 Gregory cooperated with colleagues in Paris to make simultaneous observations of an eclipse of the moon and he was able to work out the longitude for the first time. However he had already begun work on an observatory. In 1673 the university allowed Gregory to purchase instruments for the observatory, but told him he would have to make applications and organise collections for funds to build the observatory. Gregory went home to Aberdeen and took a collection outside the church doors for money to build his observatory. On 19 July 1673 Gregory wrote to Flamsteed, the Astronomer Royal, asking for advice. He then travelled to England to purchase instruments.

Gregory left St Andrews for Edinburgh in 1674. His reasons for leaving again paint a sorry picture of prejudice against the brilliant mathematician. Writing after taking up his Edinburgh chair Gregory said:-

In Edinburgh Gregory became the first person to hold the Chair of Mathematics there. He was not to hold the chair for long, however, for he died almost exactly one year after taking up the post. It was a year in which he was still very active in research in both astronomy and mathematics. On the latter topic he had become interested in the problem of solving quintic equations algebraically and made some interesting discoveries on Diophantine problems. His death came suddenly. One night he was showing the moons of Jupiter to his students with his telescope when he suffered a stroke and became blind. He died a few days later at the young age of 36. Whiteside writes in [1]:-I was ashamed to answer, the affairs of the Observatory of St Andrews were in such a bad condition, the reason of which was, a prejudice the masters of the University did take at the mathematics, because some of their scholars, finding their courses and dictats opposed by what they had studied in the mathematics, did mock at their masters, and deride some of them publicly. After this, the servants of the colleges got orders not to wait on me at my observations: my salary was also kept back from me, and scholars of most eminent rank were violently kept from me, contrary to their own and their parents wills, the masters persuading them that their brains were not able to endure it.

We have mentioned in this article many of the brilliant ideas which are due to Gregory. However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor series more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π andFor all his talent and promise of future achievement, Gregory did not live long enough to make the major discovery which would have gained him popular fame. For his reluctance to publish his "several universal methods in geometry and analysis" when he heard through Collins of Newton's own advances in calculus and infinite series, he postumously paid a heavy price ...

*e*are not the solution of algebraic equations; he knew how to express the sum of the

*n*th powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.

**Article by:** *J J O'Connor* and *E F Robertson*