Lecture 18 Barrow's proof of the fundamental theorem of calculus.

Background: Calculus between Fermat and Newton.

The historical order of the five essential steps in the development of Calculus are almost the exact opposite of the order in which we teach them::
  1. Areas of figures and volumes of solids of revolution.
  2. Tangents to curves
  3. Power Series
  4. The Fundamental Theorem of Calculus, both INT ab f'(x) dx = f(b) - f(a) and d/dx INTax f(t) dt = f(x).
  5. Limits.

We have seen how by 1650 fundamental discoveries were made in 1. and 2. by Descartes, Fermat, de Roberval and others we have not studied, including Cavalieri, Torricelli and Pascal. As for series, recall that Archimedes used a geometric series in his quadrature of the parabola. Geometric series in general were known to the Indians and other types of series were discovered by the English John Wallis (1616--1703), the Danish--English Nicolaus Mercator ( not Gerard, the cartographer) (1620--1687) and the Scots John Napier (1550--1617) and James Gregory (1638--1675). For example, the latter had a geometric derivation of the arctan series arctan x = x - x3/3 + x5/5 - ... from which pi/4 = 1 - 1/3 + 1/5 - 1/7 +...

Now we discuss the work of Isaac Barrow (1630--1677) on the FT of C. It was well--known by 1650 that the area under the graph of y=xn is 1/(n+1) xn+1, when n is a rational not -1 and that the derivative of 1/(n+1)xn+1 is xn, so both parts are valid for this case. But there was no general Theorem published before Barrow and for this reason, some scholars regard him as the Father of Calculus.

Barrow studied Mathematics and Classics at Trinity College, Cambridge taking his MA in 1652. This was a time of civil unrest in Britain, Cromwell defeated the Royalists in 1649, Charles I was beheaded in 1651. It was a period of religious intolerance and even warfare, CE v. Protestants (Puritans). The Restoration of Charles II in 1660 marked the beginning of a more tolerant period. Barrow was a Royalist and a devout CE cleric so was ousted from Cambridge in 1655, travelling and studying four years on the Continent. He became a minister and Professor of Greek at Cambridge in 1662, and Professor of Geometry in London and Lucasian Professor of Mathematics in Cambridge in 1663. He published his Geometrical Lectures in 1669, and then resigned his Chair in favour of the 26 year old Newton, becoming Royal Chaplain and later VC of Cambridge.

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The text: from Barrow's Geometrical Lectures

Let ZGE be any curve of which the axis is VD and let there be perpendicular ordinates to this axis (VZ, PG, DE) continually increasing from the initial ordinate VZ; also let VIF be a line such that, if any straight line EDF is drawn perpendicular to VD, cutting the curves in the points E, F, and VD in D, the rectangle contained by DF and a given length R is equal to the intercepted space VDEZ; also let DE:DF = R:DT, and join T and F. Then TF will touch the curve VIF. For, if any point I is taken in the line VIF (first on the side of F towards V), and if through it IG is drawn parallel to VZ, and IL is parallel to VD, cutting the given lines as shown in the figure; then LF:LK = DF:DT = DE:R, or R x LF = LK x DE.

But, from the stated nature of the lines DF, LK, we have R x LF = area PDEG: therefore LK x DE = area PDEG < DP x DE; hence LK < DP < LI.

Again, if the point I is taken on the other side of F, and the same construction is made as before, plainly it can be easily shown that LK > DP > LI.

From which it is quite clear that the whole of the line TKF lies within or below the curve VIFI.

Other things remaining the same, if the ordinates VZ, PG, DE, continually decrease, the same conclusion is attained by a similar argument; only one distinction occurs, namely, in this case, contrary to the other, the curve VIF is concave to the axis VD.

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Explanation

The extract is from the Geometrical Lectures, written in Latin, and concerns part of the FT of C, namely the slope at x of the curve representing INTax f(t) dt is f(x) .

The curve f(t) is ZGE, increasing , though drawn below the axis. The curve g(x)= INTax f(t) dt is VIF. In order that an ordinate g(x) on this curve represent an area INTax f(t) dt , Barrow introduces the fixed length R such that R x g(x) is the area, i.e. R is a device to keep the dimensions consistent. We could take R = 1.

He then wishes to prove that the subtangent t(x) to g(x) at (x,g(x)) is Rg(x)/f(x) , or as we would say, g'(x) = g(x)/t(x) = f(x)/R .

He does this by showing that the line through (x,g(x)) whose slope DF/DT is DE/R = f(x)/R is in fact the tangent, because it touches the curve at (x,g(x)) and lies beneath it for all y < x and also for all y > x.

In a later section of the Lectures, Barrow also proves the second half of the FT of C.

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For a biography of Barrow, and a glimpse of the original version of our extract.

For more on the history of the Fundamental Theorem of Calculus.

To return to Table of Contents.

For next Lecture .

Last update: 22 August, 2000

Author: Phill Schultz, schultz@maths.uwa.edu.au