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Topic: [HM] source for a story
Replies: 40   Last Post: Jul 20, 2004 4:32 PM

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James A Landau

Posts: 217
Registered: 12/3/04
Re: [HM] Granville
Posted: Jul 10, 2004 7:08 PM
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I have in front of me copies of the 1904 edition of Granville and the 1941
edition of Granville, Smith, and Longley

The 1904 edition, as far as I can find, does NOT use the name "Fundamental
Theorem of Calculus". The 1941 edition has (page 254, Cahpter XV, section 156)
"The Fundamental Theorem of integral calculus."

The 1904 edition (chapter XXIX, section 209, page 356) reads (I have
condensed the quote somewhat so as to use only ASCII characters)
<quote>
Since (indefinite integral of) phi (x) dx exists, denote it by f(x) + C,
(B) u = f(x) + C

We may determine C if we know the value of u for some value of x. If we
agree to reckon the area from the axis of y, ie.e when (C) x = a, u = area OCDG [O
the origin, G is point (0, f(0), C is (a, 0), and D is (a, f(a)], and when x
= b, u = area OEFG, etc.,
it follows that if (D) x = 0, then u = 0.
Substituting (D) in (B), we get u = f(0) + C, or C = - f(0).

Hence from (B) we obtain (E) u = f(x) - f(0)

To find the area between the ordinates CD and EF, substitute the values (C)
in (e), giving
(F) area OCDG = f(a) - f(0)
(G) area OEFG = f(b) - f(0)
Subtracting (F) from (G),
(H) area CEFD = f(b) - f(a)
<end quote>

At the top of page 357, this material is stated as a theorem: "The difference
of the values of integral ydx for x = a and x = b gives the area bounded y
the curve whose 0ordinate is y, the axis of X, and the ordinates corresponding
to x=a and x = b."

Then in chapter XXX, section 229, page 367, the book gives a more detailed
proof of this theorem, dividing the curve into intervals and using what
Granville calls the "Theorem of an Value" to come to the conclusion "(page 369) "This
exhibits our definite integral as the limit of a sum of differential
expressions.". Since Granville says (page 369) let us choose within the interval [a,
b], n-1 abscissas, x1, x2,...xn-1, in any manner whatever" which I think means
that he is defining his definite integral as a Riemann integral.

Then, in small print on page 370, "In order to replace the intuitional point
of view that we have so far adopted in the text by a rigour and general
analytical proof..." followed by an argument that includes the statement "it is
always possible [dagger] to make all these differences less in numerical value
than any assignable positive number epsilon, however small..." which, if I
correctly understand Granville's wording, means that he his using an argument
based on uniform convergence. "dagger" is a footnote reading ""That such is the
case is shown in advanced works on the Calculus."

The 1941 edition uses the same approaches on defining the definite integral
and FTC and gas most of the same wording. Oddly, the caveat "In order to
replace the intuitional point of view that we have so far adopted in the text by a
rigour and general analytical proof..." does not appear in the 1941 edition.

I will be happy to supply additional material from either edition on request.

A quick look at the table of contents of the two editions shows that both
have much the same material, but frequently in a different order (e.g. the 1904
edition has partial derivatives before integration, the 1941 edition introduces
them three chapters from the end of the book.) The two editions are roughly
the same length. The 1941 edition has material on hyperbolic functions,
centroids, and applications of partial derivatives that are not in the 1904 edition.

- James A. Landau





Date Subject Author
5/13/04
Read [HM] source for a story
Fernando Q. Gouvea
5/13/04
Read Re: [HM] source for a story
David Derbes
6/30/04
Read [HM] FTC
David M. Bressoud
6/30/04
Read Re: [HM] FTC
Mike Robison
6/30/04
Read Re: [HM] FTC
Martin Davis
6/30/04
Read Re: [HM] FTC
Thierry Guitard
6/30/04
Read Re: [HM] FTC
Fernando Q. Gouvea
7/1/04
Read Re: [HM] FTC
Udai Venedem
7/1/04
Read Re: [HM] FTC
William C Waterhouse
7/2/04
Read Re: [HM] FTC
Gordon Fisher
7/6/04
Read [HM] Granville
Roger Cooke
7/6/04
Read Re: [HM] Granville
Gordon Fisher
7/9/04
Read [HM] Teaching epsilonics to engineers
Roger Cooke
7/10/04
Read Re: [HM] Teaching epsilonics to engineers
Robert (Bob) Eldon Taylor
7/11/04
Read Re: [HM] Teaching epsilonics to engineers
Roger Cooke
7/12/04
Read [HM] What language did the Bernoullis speak at home?
Roger Cooke
7/14/04
Read Re: [HM] What language did the Bernoullis speak at home?
Siegmund.Probst
7/16/04
Read Re: [HM] What language did the Bernoullis speak at home?
Hans Samelson
7/14/04
Read Re: [HM] What language did the Bernoullis speak at home?
Peter Flor
7/12/04
Read Re: [HM] Teaching epsilonics to engineers
Gordon Fisher
7/11/04
Read [HM] A philological question on Blossoms
Roger Cooke
7/15/04
Read Re: [HM] A philological question on Blossoms
Daniel Otero
7/20/04
Read Re: [HM] A philological question on Blossoms
William C Waterhouse
7/12/04
Read Re: [HM] Teaching epsilonics to engineers
Diana Kornbrot
7/13/04
Read Re: [HM] Teaching epsilonics to engineers
John Harper
7/12/04
Read Re: [HM] Teaching epsilonics to engineers
Don Cook
7/13/04
Read Re: [HM] Teaching epsilonics to engineers
Roger Cooke
7/14/04
Read Re: [HM] Teaching epsilonics to engineers
Gordon Fisher
7/16/04
Read [HM] Axiomatics
Samuel S. Kutler
7/16/04
Read Re: [HM] Axiomatics
Elena A Marchisotto
7/16/04
Read Re: [HM] Teaching epsilonics to engineers
James A Landau
7/9/04
Read Re: [HM] Granville
Robert L. Knighten
7/6/04
Read [HM] Granville
Gordon Fisher
7/9/04
Read Re: [HM] Granville
David Derbes
7/9/04
Read Re: [HM] Granville
Lindsey, Dr. Charles
7/10/04
Read Re: [HM] Granville
James A Landau
7/6/04
Read Re: [HM] FTC
Bruce S. Burdick
7/6/04
Read [HM] Workshop on the criticism of diagrams in mathematical texts
Veronica Gavagna
7/7/04
Read Re: [HM] Workshop on the criticism of diagrams in mathematical texts
Marcus Barnes
7/6/04
Read Re: [HM] FTC
James Buddenhagen
7/7/04
Read Re: [HM] FTC
Thierry Guitard

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