For a long time the light of James Gregory did not shine as brightly as did that of John Wallis, Isaac Barrow and Isaac Newton, the other three great British mathematicians of the seventeenth century. Only recently, through the endeavors of several Scottish mathematicians, especially E. T. Whittaker, G. A. Gibson and H. W. Turnbull, Gregory's genius is revealed and fills with admiration all those interested in the development of modern mathematics.

The James Gregory Tercentenary Memorial Volume, edited by H. W. Turnbull [1], contains Gregory's momentous scientific correspondence, mostly with J. Collins. An extremely important supplement is the large number of Gregory's hitherto unpublished notes, recording his mathematical ideas and calculations. These notes were found in a collection of documents in the University of St. Andrews Library, written on the blank spaces of letters to Gregory. This material affords the possibility of studying his achievements and ideas.

In this paper we shall discuss Gregory's expansions of general and particular functions into series. In addition, we shall exhibit the ideas which are set forth in his first mathematical publication Vera circuli et hyperbolae quadratura [2]. These ideas are concerned, to some extent, with the associated problem of constructing by certain limiting processes the functions which measure the areas of circles and conics.

The “Taylor's series”

In a letter of February 15, 1671 to J. Collins (see Memorial [1], pp. 170 ff.) Gregory gives the power series for seven important functions, each with 5 or 6 terms.