**James Gregory** was a Scottish scientist and first Regius Professor of Mathematics at St Andrews who described the first practical reflecting telescope. He worked on using infinite convergent series to find the areas of the circle and hyperbola.

- 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.
- James was given Euclid's Elements to study and he found this quite an easy task.
- Gregory's health was poor in his youth.
- Gregory began to study optics and the construction of telescopes.
- 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.
- Next Gregory gives details of his invention of a reflecting telescope.
- In 1663 Gregory went to London.
- One of Gregory's aims was to have Optica Promota published and he achieved this.
- 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.
- 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.
- Moray was to play a major role in Gregory's career somewhat later.
- In 1664 Gregory went to Italy.
- 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.
- 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 was really trying to prove that π and e are transcendental but Gregory's arguments contain a subtle error.
- 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.
- 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.
- 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.
- 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.
- We can now be certain that during the summer of 1668 Gregory was completely familiar with the series expansions of sin, cos and tan.
- 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.
- We have already mentioned that Robert Moray was a member of the Royal Society with whom Gregory was friendly.
- Gregory arrived in St Andrews late in 1668.
- Gregory found that St Andrews was of classical outlook where the latest mathematical work was totally unknown.
- However Gregory was to carry out much important mathematical and astronomical work during his six years in the Regius Chair.
- Gregory preserved all Collins's letters, writing notes of his own on the backs of Collins's letters.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Gregory left St Andrews for Edinburgh in 1674.
- In Edinburgh Gregory became the first person to hold the Chair of Mathematics there.
- 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 and eee are not the solution of algebraic equations; he knew how to express the sum of the nnnth 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.

Born November 1638, Drumoak (near Aberdeen), Scotland. Died October 1675, Edinburgh, Scotland.

View full biography at MacTutor

Ancient Greek, Ancient Indian, Astronomy, Geometry, Origin Scotland, Number Theory, Physics, Puzzles And Problems, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive