Recent examination of original and mostly unpublished manuscripts and letters has thrown new light upon the early developments of the Calculus by James Gregory (1638-1675) and Isaac Newton (1642-1727).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
y2(l + (dy/dx)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 David Gregory (1661-1710) left unpublished notes including those of a projected History of Fluxions, and of discussions with Newton, 1690 onwards, on mathematical and physical problems.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
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