James gregory 1638 biography examples
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These expansions highlighted the potential of series to represent entire functions analytically, influencing the formalization of power series in calculus.[1]In his work on logarithms, Gregory employed and extended the series for the natural logarithm,ln(1+x)=x−2x2+3x3−4x4+⋯for ∣x∣<1, originally discovered by Nicolaus Mercator in 1668.
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. 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.
On 19 July 1673 Gregory wrote to Flamsteed, the Astronomer Royal, asking for advice.
James Gregory
James Gregory (1638-1675) was a Scottish mathematician, astronomer, and first Regius Professor of Mathematics at the University of St Andrews. Semesterber.5(1956), 143-146.
The tube of the Gregorian telescope is thus shorter than the sum of the focal lengths of the two mirrors. He also worked to find the areas of the circle and hyperbola using a modification of the method of Archimedes (c.211BCE).
Gregory was elected to the Royal Society of London before travelling to St Andrews and there starting his family. 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.
Gregory arrived in St Andrews late in 1668. Mag.
Gregory hung his pendulum clock on the wall beside the same window. I would gladly hear Mr Newton's thoughts of it. 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 integrated this series into his quadrature techniques and used it to approximate logarithmic values, comparing partial sums to geometric progressions for convergenceanalysis.
However, Huygens' main mathematical objection to Gregory's proof is a valid one. It was reprinted in 1668 with an appendix, Geometriae Pars, in which Gregory explained how the volumes of solids of revolution could be determined.
Gregorian telescope
Main article: Gregorian telescope
In his 1663 Optica Promota, James Gregory described his reflecting telescope which has come to be known by his name, the Gregorian telescope.
Reprinted in: Correspondence of Scientific Men of the Seventeenth Century...., ed. This technique paralleled later observations and demonstrated his innovative approach to quantitative astronomy.
There, he pursued a broad curriculum encompassing classics, philosophy, and mathematics, drawing on foundational knowledge in geometry imparted by his mother and elder brother David during his early years. 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 e are not the solution of algebraic equations; he knew how to express the sum of the nth 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.
- D T Whiteside, Biography in Dictionary of Scientific Biography(New York 1970-1990).
The feather became the first diffraction grating but again Gregory's respect for Newton prevented him going further with this work. Essentially Newton and Gregory were working out the basic ideas of the calculus at the same time, as, of course, were other mathematicians. These interactions exposed him to cutting-edge Italian mathematical traditions and unpublished works, profoundly shaping his analytical methods and leading to the publication of two influential treatises: Vera circuli et hyperbolae quadratura in 1667 and Geometriae pars universalis in 1668.[1][2][4]Gregory returned to Scotland in late 1668, carrying the intellectual influences of his European travels, including rigorous geometric approaches and access to continental unpublished manuscripts that enriched Scottish mathematical discourse.
These expansions were derived independently and communicated through letters, predating similar discoveries by other mathematicians.[1]Gregory developed series for sine and cosine by summer 1668, serving as precursors to the general Taylor series framework he outlined in 1671.