James Gregory

Scottish astronomer and mathematician

• Born: November 1, 1638
• Birthplace: Drumoak, Aberdeenshire, Scotland
• Died: October 1, 1675
• Place of death: Edinburgh, Scotland

Gregory designed the first practical reflecting telescope and proposed utilizing light intensity to estimate stellar distances. He formulated methods anticipating the discovery of calculus, developed infinite series representations for various trigonometric functions, and was the first person to propose and prove the rudimentary theoretical proposition today known as the fundamental theorem of calculus.

Early Life

James Gregory was born in the Manse of Drumoak, 9 miles (15 kilometers) west of Aberdeen, to the Rev. John Gregory, an Episcopalian clergyman, and Janet Anderson. The youngest of three children, James was often sick as a child. Perhaps for this reason, his mother, an intelligent, educated woman, taught him mathematics and geometry. Following their father’s death in 1650, James’s twenty-three-year-old brother, David Gregory (an amateur mathematician), gave James a copy of Greek mathematician Euclid’s Elementa (c. 300 b.c.e.; Elements, 1570) to encourage his latent talent. Easily mastering the material, James was sent on to the Aberdeen Grammar School. He then proceeded to Marischal College, Aberdeen, where he focused his studies on astronomy and mathematical optics.

After graduating in 1657, Gregory devoted his energy to studying optics and telescope construction. Encouraged by his brother David, he wrote a treatise summarizing five years of original research. Titled Optica Promota (1663, the advance of optics), this work proved theorems on the reflection and refraction of light, presented propositions on mathematical astronomy, and discussed photometric methods to estimate stellar distances.

The book’s greatest contribution, however, was an exposition of the first practical reflecting telescope utilizing a concave mirror to focus light. Gregory’s innovation employed a small concave mirror near the top of the telescope to reflect light from the focusing mirror back down the telescope tube through a small aperture in the center of the primary mirror, where it formed an image that could be examined with an eyepiece. Gregory’s design had two advantages over refracting telescopes: The tube was more compact and the color distortion (chromatic aberration) introduced by an objective lens was nonexistent. Because Gregory did not possess the skill to grind and polish a mirror to the correct shape, he abandoned his concept in 1664 and journeyed to Padua, Italy, devoting himself exclusively to mathematical studies.

Life’s Work

Enrolling at the University of Padua in 1664, Gregory spent the next four years studying geometry, mechanics, and mathematical astronomy under the tutelage of Stefano degli Angeli. While studying at Padua, Gregory produced his first mathematical treatise, Vera Circuli et Hyperbolae Quadratura (1667; the true squaring of the circle and of the hyperbola). This text was the first to distinguish between an infinite series of summed terms that converge and those that diverge. (When all the terms of a converging series are added, a finite limit is approached, unlike a diverging series, whose sum approaches infinity.) Gregory used convergent infinite series to calculate, respectively, the areas of circles and hyperbolas.

The succeeding year saw the publication of a second insightful treatise, even more general and abstract than the first. Titled Geometriae Pars Universalis (1668; the universal part of geometry), this opus presented rules for finding the areas of curves and the volumes of their solids of revolution (that is, the volume generated when a two-dimensional curve is rotated about an axis). In the process of producing this work, Gregory formulated two key aspects of calculus , differentiation and integration, in a consistent systematic manner. Although Sir Isaac Newton has been given priority for inventing the calculus, Gregory and several other mathematicians were working out the ideas independently at about the same time.

Gregory returned to England in the spring of 1668. Based on his books, Gregory had acquired a sufficient commendatory reputation in mathematics that he was elected a fellow of London’s prestigious Royal Society soon after his return. Through Gregory’s connections in the Royal Society, King Charles II was persuaded to create an endowed chair of mathematics at the University of Saint Andrews, Scotland, to provide Gregory a professorship from which he could continue his distinguished mathematical research. His reputation now secured, Gregory took up residence at Saint Andrews late in 1668. The succeeding year, he married a young widow, Mary Jamesome, who would bear him two daughters and a son.

During his tenure at Saint Andrews, Gregory carried out much important mathematical and astronomical work. In his Exercitationes geometricae (1669; geometrical exercises), Gregory developed an analytical method of drawing tangents to curves (in the parlance of calculus, this would become known as differentiation). He kept in touch with current research by corresponding with other members of the Royal Society, including Newton, whom Gregory greatly admired. Based on this correspondence, Gregory incorporated many of Newton’s ideas into his own teaching, even though these concepts were considered quite controversial at the time.

Due to an earlier controversy with Christiaan Huygens , who falsely claimed authorship of sections of Vera circuli et Hyperbolae Quadratura, Gregory was reluctant to publish much of his work or disclose the methods by which he made discoveries. Consequently, it was not until the James Gregory Tercentenary in 1938, when his papers were exhumed from the archives of Saint Andrews’s library, that the full extent of his brilliance was realized. For example, had discovered the principle now known as Taylor’s theorem in February, 1671; it was not published by Brook Taylor until 1715.

Gregory made another important scientific discovery when he utilized the feather of a sea bird to observe the diffraction of light, a phenomenon explicable only if light consisted of waves. Because Newton believed light had a corpuscular nature and Gregory had an enormous respect for Newton, he pursued this concept no further. Consequently, Gregory received only a fraction of the credit he deserved during his lifetime and even less during the ensuing centuries. The true magnitude of his achievements was not acknowledged until the 1930’s, when, in the wake of the tercentenary, his notes and correspondence were examined and published by H. W. Turnbull as James Gregory: Tercentenary Memorial Volume (1939).

Increasing prejudice against the brilliant mathematician by Saint Andrews’s classically oriented faculty and administration caused Gregory to resign his position at the end of the 1674 spring term. Eager to acquire this young, productive scholar, Edinburgh University created a new position for him, their first chair of mathematics. Unfortunately, this was a post he was to occupy for only one year. In October, 1675, while observing the moons of Jupiter through a telescope, Gregory suffered a blinding stroke, dying several days later at the age of thirty-six.

Significance

Gregory was one of the most important mathematicians of the seventeenth century, significant especially in the steps that led to the calculus. Unfortunately, like so many other scientific luminaries of the seventeenth century, his brilliance was eclipsed by that of Isaac Newton. His lack of historical appreciation was further exacerbated by his reluctance to publish his methods and his relatively short life. Some of his remarkable contributions, only recently brought to light, include his discovery of the general binomial theorem several years before Newton and his exposition of so-called Taylor expansions forty years before Taylor. He studied infinite series and elucidated one of the earliest examples of a test for a series’s convergence.

In calculus, Gregory’s definition of the integral was well formulated in a completely general form, and he had acquired a profound understanding of the various solutions possible for differential equations. He was the first mathematician who attempted to prove that π and e are irrational numbers, and he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of their coefficients. His correspondence also suggests that at the time of his death he had begun to realize that algebraic equations of degree greater than four could not be solved by equations in closed form.

Although possessing an enormous talent for mathematics and exhibiting tremendous promise for outstanding future accomplishments, Gregory’s relatively short life precluded him from realizing any major discoveries, publishing them, and receiving the critical acclaim he most definitely deserved. During his last years of life, Gregory was reluctant to publish important results, and he was reticent to engage in controversy or proprietary arguments with Newton once he heard of Newton’s advances in calculus and infinite series. This reluctance posthumously exacted a heavy toll upon his place in history.

Bibliography

Dehn, M., and E. Hellinger. “Certain Mathematical Achievements of James Gregory.” American Mathematical Monthly 50 (1943): 149-163. This article discusses Gregory’s anticipation of important mathematical discoveries in number theory, differential calculus, and infinite series.

Malet, A. “James Gregorie on Tangents and the ’Taylor’ Rule for Series Expansions.” Archives for History of Exact Science 46 (1993-1994): 97-137. Explains how Gregory’s tangent rule is essentially equivalent to differentiation and how Gregory proposed, but never published, the famed Taylor expansion decades before Taylor.

Scriba, C. J. “Gregory’s Converging Double Sequence: A New Look at the Controversy Between Huygens and Gregory Over the ’Analytical Quadrature of the Circle.’” Historia Math 10, no. 3 (1983): 274-285. A critical account of Huygens’s attack on Gregory’s Vera circuli, as well as Gregory’s rebuttal, which proves that Huygen’s aggressive assault was unfounded and unnecessary.

Simpson, A. D. C. “James Gregory and the Reflecting Telescope.” Journal of the History of Astronomy. 23, no. 2 (1992): 77-92. A brief account of Gregory’s design for a practical telescope and his futile search for a London optician who could correctly grind and polish the mirrors.

Turnbull, H. W. “James Gregory.” In James Gregory Tercentenary Memorial Volume, edited by H. W. Turnbull. London: G. Bell & Sons, 1939. In addition to articles discussing Gregory’s major works and the Gregory/Huygens controversy, this volume contains copies of Gregory’s letters and posthumous manuscripts.