
Re: [HM] Granville
Posted:
Jul 10, 2004 7:08 PM


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], n1 abscissas, x1, x2,...xn1, 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

