History of Calculus
Thom Jameson
1. How Newton and Leibniz independently invent calculus.
2. What were the controversies that followed.
3. How did the two theories continue to evolve? Did one theory win over the other?
Can you please recommend any.
Dave L. Renfro
The following 3 books would be my top choices for
what you're looking for:
Margaret E. Baron, "The Origins of the Infinitesimal Calculus",
2004. [Dover reprint of the 1969 edition.]
http://store.doverpublications.com/0486495442.html
http://tinyurl.com/y3hknw [amazon.com information]
Carl B. Boyer, "The History of Calculus and Its Conceptual
Development", 1949/1959. [Dover reprint of the 1949 edition.]
http://store.doverpublications.com/0486605094.html
http://tinyurl.com/y973wc [amazon.com information]
C. H. Edwards, "The Historical Development of the Calculus", 1994.
http://tinyurl.com/y5bfck [amazon.com information]
See also the first 2 chapters of the following, which are
excellent for what you're asking about:
Ivor Grattan-Guinness (editor), "From the Calculus to
Set Theory, 1630-1910: An Introductory History", 2000.
[Paperback reprint of the 1980 edition.]
http://tinyurl.com/ufxul [amazon.com information]
Dave L. Renfro
Thom Jameson
Dave L. Renfro
The book I cited by Carl B. Boyer might be a little
easier to follow. However, I haven't looked at Baron's
book in a while, so I'm not sure about this. I do know
that I've always found it very difficult to read about
the history of various math ideas. Mathematical historians
often don't rephrase things very well in modern language
(in my opinion), and I sometimes find it difficult to
extract from their prose *precisely* _who_ knew _what_
and _when_, or at least a clear distinction of what we
know today about these issues and what we don't know
today about these issues. I know the overall evolution
of ideas is more important, and often many different
people contribute in various ways to something that
was only later recognized as significant by someone,
but honestly, I often feel like I'm pulling teeth
when I go to a history of mathematics text for what
I would consider to be a fairly basic issue. For example,
who was the first to consider the possibility of infinitely
many oscillations in a bounded interval and to what end?
(I think it was Cauchy, 1817 or 1818, and it was small
positive powers of x times sin(1/x) to, I think,
illustrate his definition of continuity.) Another
example: Who/when was the first to consider the
possibility that a derivative might not be continuous,
and who/when actually came up with such an example?
(I don't know about the first question, but I think
Darboux in January 1874, and published in 1875, was
the first to come up with an example, the still
standard (x^2)*sin(1/x) example.)
In your case, I'd strongly advise you to go to a
university library, look up the library call numbers
of the four books I gave earlier (if they have them),
and go to where those books are on the shelves and
look for other books where those are located.
Also, the references given in the paper I cite
in the following post will lead you to quite a few
articles about the development of calculus. If you're
interested, I could perhaps post some of the article
titles listed in that reference, but I can't do this
right now because the paper is at home and I'm not.
http://mathforum.org/kb/message.jspa?messageID=4702130
Dave L. Renfro
Dave L. Renfro
> Also, the references given in the paper I cite
> in the following post will lead you to quite a few
> articles about the development of calculus. If you're
> interested, I could perhaps post some of the article
> titles listed in that reference, but I can't do this
> right now because the paper is at home and I'm not.
>
> http://mathforum.org/kb/message.jspa?messageID=4702130
I got the article out that I was talking about. Using this
article, and a little bit of googling, I've collected below
some references that are related to the Newton/Leibniz era
of calculus. They are listed by journal (and chronologically
within each journal's list) for convenience in looking them
up in a library, except for a final list of "Other Journals"
with some relevant papers from more research oriented journals.
The following internet web pages should also be of interest.
George Berkeley's influential 1734 essay "The Analyst"
http://www.maths.tcd.ie/pub/HistMath/People/Berkeley/AnalCont.html
Bjørn Smestad, "Foundations for fluxions", Cand. Scient Thesis
in Mathematics, Department of Mathematics, University of Oslo,
1995.
http://home.hio.no/~bjorsme/hovedoppg.HTM
Joel A. Tropp, "Infinitesimals: History & Application",
Plan II Honors Thesis, University of Texas at Austin,
May 1999, 87 pages.
http://www-personal.umich.edu/~jtropp/papers/Tro99-Infinitesimals.pdf
http://tinyurl.com/zab7v [tinyrul equivalent]
http://en.wikipedia.org/wiki/Newton_v._Leibniz_calculus_controversy
http://en.wikipedia.org/wiki/History_of_calculus
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html
*************** AMERICAN MATHEMATICAL MONTHLY ***************
Florian Cajori, "Discussion of fluxions: from Berkeley to
Woodhouse", American Mathematical Monthly 24 (1917), 145-154.
Griffith Conrad Evans, "Cavalieri's theorem in his own words",
American Mathematical Monthly 24 (1917), 447-451.
Florian Cajori, "Who was the first inventor of calculus",
American Mathematical Monthly 26 (1919), 15-20.
Florian Cajori, "Grafting of the theory of limits on the
calculus of Leibniz", American Mathematical Monthly 30 (1923),
223-234.
J. A. Garrett and Frank Lynwood Wren, "The development of the
fundamental concepts of infinitesimal analysis", American
Mathematical Monthly 40 (1933), 269-281.
Arthur Rosenthal, "The history of calculus", American Mathematical
Monthly 58 (1951), 75-86.
Julian Lowell Coolidge, "The story of tangents", American
Mathematical Monthly 58 (1951), 449-462.
Sydney Henry Gould, "The method of Archimedes", American
Mathematical Monthly 62 (1955), 473-476.
******************** MATHEMATICS MAGAZINE *******************
Judith V. Grabiner, "The changing concept of change: the
derivative from Fermat to Weierstrass", Mathematics Magazine
56 (1983), 195-206.
**************** COLLEGE MATHEMATICS JOURNAL ****************
Carl B. Boyer, "The history of the calculus", (Two-Year) College
Mathematics Journal 1 #1 (Spring 1970), 60-86.
V. Frederick Rickey, "Isaac Newton: man, myth, and mathematics",
College Mathematics Journal 18 (1987), 362-389.
******************** SCRIPTA MATHEMATICA ********************
Lao G. Simons, "The adoption of the method of fluxions in
American Schools", Scripta Mathematica 4 (1936), 207-219.
Carl B. Boyer, "Cavalieri, limits and discarded infinitesimals",
Scripta Mathematica 8 (1941), 79-91.
Russell W. Cowan and Bernard C. Weber, "Fermat's contributions
to the development of the differential calculus", Scripta
Mathematica 13 (1947), 123-127.
******************* MATHEMATICAL GAZETTE *******************
W. D. Evans, "Berkeley and Newton", Mathematical Gazette
7 (1913-14), 418-421.
******************** MATHEMATICS TEACHER ********************
G. H. Graves, "Development of the fundamental ideas of the
differential calculus", Mathematics Teacher 3 (1910), 82-89.
Carl B. Boyer, "The first calculus textbooks", Mathematics
Teacher 39 (1946), 159-167.
Carl B. Boyer, "History of the derivative and integral of
the sine", Mathematics Teacher 40 (1947), 267-275.
Carl B. Boyer, "The quadrature of the parabola: an ancient
theorem in modern form", Mathematics Teacher 47 (1954), 36-37.
Dorothy V. Schrader, "The Newton Leibniz controversy concerning
the discovery of the calculus", Mathematics Teacher 55 (1962),
385-396.
*********************** THE PENTAGON ************************
M. Thomas, "The Newton-Leibniz controversy", The Pentagon
3 (1943), 28-36.
*********************** OTHER JOURNALS ***********************
Florian Cajori, "The spread of Newtonian and Leibnizian notations
of the calculus", Bulletin of the American Mathematical Society
27 (1921), 453-458.
J. M. Child, "Barrow, Newton and Leibniz, in their relation
to the discovery of the calculus", Science Progress 25 #98
(1930), 295-307.
Augustus De Morgan, "On the early history of infinitesimals
in England", Philosophical Magazine (4) 4 (1852), 321-330.
G. A. Gibson, "Berkeley's Analyst and its critics: An episode
in the development of the of the doctrine of limits", Bibliotheca
Mathematica (N.S.) 13 (1899), 65-70.
Philip Kitcher, "Fluxions, limits, and infinite littlenesse:
A study of Newton's presentation of the calculus", Isis
64 (1973), 33-49.
Christoph J. Scriba1, "The inverse method of tangents: A
dialogue between Leibniz and Newton (1675-1677)", Archive for
History of Exact Sciences 2 (1964), 113-137.
G. C. Smith, "Thomas Bayes and fluxions", Historia Mathematica
7 (1980), 379-388.
Dave L. Renfro