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

**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).

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

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.

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.

He combines both the indivisibility and infinitesimal ideas.

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

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.

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**

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

Since

and

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,

**SUMMARY**

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

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.

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

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

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

DID BARROW INVENT THE CALCULUS? NO.

Thu Apr 10 06:48:29 CDT 1997