# H W Turnbull: *Scottish Contribution to the Calculus*

An International Congress of Mathematicians was held in Cambridge, Massachusetts, USA from 30 August to 6 September 1950. Several lectures given in the "History" part of Section VII "History and Education". H W Turnbull gave a lecture in "History" on

*The Scottish Contribution to the Early History of the Calculus.*Here are some notes which indicate the content of the lecture:-### THE SCOTTISH CONTRIBUTION TO THE EARLY HISTORY OF THE CALCULUS

**H W Turnbull**

Recent examination of original and mostly unpublished manuscripts and letters has thrown new light upon the early developments of the Calculus by James Gregory (1638-1675) and Isaac Newton (1642-1727). Interpolation formulae involving successive order of finite differences as well as the power series, involving successive derivatives and found by Taylor and Maclaurin, were used over forty years earlier (1670-1671) by Gregory. The differential equation

$y^{2}(l + (<f+>\large\frac{dy}{dx}\normalsize </f>)^{2}) = f(x)$

in geometrical guise was discussed by Gregory and Barrow. There are examples of independent and almost simultaneous discoveries by Gregory and Newton. Light is thrown on the problem of dating accurately certain results of Newton which were published only many years later. David Gregory (1661-1710) left unpublished notes including those of a projected

*History of Fluxions*, and of discussions with Newton, 1690 onwards, on mathematical and physical problems. The notes provide evidence of work by Hudde at Amsterdam prior to 1660 on the logarithmic series, antedating Newton and Mercator. Letters and manuscripts of Colin Campbell bridging the years between J Gregory and Maclaurin give a vivid picture of scientific activity in Edinburgh and in the Highlands.

University of St Andrews,

St Andrews, Scotland

Last Updated March 2006