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April 10, 1997

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Early Calculus -- III

Isaac Barrow
Born: Oct 1630 in London, England
Died: 4 May 1677 in London, England

tex2html_wrap_inline126    Issac Barrow (1630-1677) was the first Lucasian Professor of Mathematics (1663) at Cambridge. He probably learned much of his mathematics during the 3 years he travelled abroad in Europe (Paris, Florence, Smyrna, Constantinople, and Venice, Germany, Netherlands).

tex2html_wrap_inline126    His interests were very broad, including Greek, religion and mathematics.

tex2html_wrap_inline126 As a professor, he was required to give lectures and in 1664-65 he gave a series of lectures which Newton almost certainly attended. The 64-65 lectures dealt with the general concepts of space, time and motion. Barrow had by this time already begun to consider the study of curves as the paths of moving points.

tex2html_wrap_inline126    His book Lectiones geometricae dates from this period, at least the first five lectures anyway. Some of the interesting points are the philosophical foundations for his ideas of Time and Motion and the geometric representation of such magnitude àla Oresme and Galileo.

tex2html_wrap_inline126    He combines both the indivisibility and infinitesimal ideas.

tex2html_wrap_inline126    Barrow covers much the same material as Gregory but does so more deeply in both foundations and development.

tex2html_wrap_inline126    He relinquished the Lucasian chair to Issac Newton in 1669 and was appointed Doctor of Divinity by Royal mandate in 1670 and Master of Trinity College in 1672.

tex2html_wrap_inline126 Barrow's lectures for the years 1664 to 1666 were only published in 1683 after his death. His Lectiones Opticae and Lectiones Geometricae were published in 1669 and 1670 respectively with Newton assisting in their preparation.

Barrows's Fundamental Theorem


tex2html_wrap_inline126    Barrow's fundamental theorem appears Lecture X, Proposition 11. From the diagram, the curve AIF represents the area under the curve ZGEG. Thus we have






it follows that


Thus, if a line FT be drawn through F, meeting the axis AD at T and such that


and if IL meets FT in K, then


and therefore


Hence K lies below the curve. To complete the proof, it is necessary to reason for points G,I' on the other side of E,F, where in the slope inequality is reversed. This proves that TF is tangent to AIF at D and has slope equal to DE. In modern terms,



tex2html_wrap_inline126    Barrow's Geometrical Lectures should be viewed as the culmination of all the 17 tex2html_wrap_inline184 century geometrical investigations leading to the calculus.

tex2html_wrap_inline126    It is the most systematic - detailed treatment of tangents arcs, areas, etc. In the hands of Newton and Leibnitz, this led to the invention of the calculus.

tex2html_wrap_inline126    Barrow integrated the concepts of time and motion with those of space suggested by Torricelli, Galileo and Roberval.

tex2html_wrap_inline126    Barrow was a skillful geometer. His thinking was geometrical. He gave little attention to analytical procedure or problem solving.

tex2html_wrap_inline126    He was somewhat disappointed by the lack of impact of his work.

tex2html_wrap_inline126 DID BARROW INVENT THE CALCULUS? NO.

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Don Allen
Thu Apr 10 06:48:29 CDT 1997