Dear Professor Cabillon
I seek your help with a historical question, and if you are not able to
help me directly perhaps you would be willing to try my question on your
email list, with a request to respond to me personally.
I seek access to the text:
A.de Sarasa (1649) Solutio problematis a R.P.Marino Mersenno minimo
propositi, I + I Meursios, Antwerp, OR to a French translation by
J.Dhombres under the title
Une algebre de raison au XVII siecle:la quadrature de l'hyperbole par
Gregoire de Saint-Vincent.
The reason for this enquiry is the difference between the description of de
Sarasa's work in recent writing, compared with descriptions before 1920.
In O.Toeplitz, The Calculus a genetic approach, in C.H.Edwards, The
historical development of the calculus and in V.Katz, A history of
mathematics, de Sarasa is credited with showing that the area under a
rectangular hyperbola from a to b is equal to the area from ta to tb, and
the consequent deduction that area(1, a) + area(1, b) = area(1, ab).
However earlier sources; Kaestner (1799) Geschichte der Mathematik,
Montucla (c.1800) Histoire des mathematiques and Cantor (1913) Vorlesungen
ueber Geschichte der Mathematik, only mention de Sarasa's attempt to solve
Mersenne's problem using a hyperbola. Mersenne had asked whether if three
numbers were given, and the logarithms of two are known, the logarithm of
the third could be calculated. In their discussion of de Sarasa, these
older authors do not give a special status to the number 1, nor do they
mention the rule log ab = log a + log b. My conjecture is that "logarithm"
to Mersenne and de Sarasa meant what it did to Napier and Burgi, namely the
matching of a geometric progression to an arithmetic progression. The terms
of the arithmetic progression being the logarithms of the terms of the
geometric progression. It is in this context that Gregory of St Vincent's
proof that a geometric progression along one asymptote of a hyperbola gave
rise to strips of equal area parallel to the other asymptote, seemed to
provide a solution.
Anyone who has had access to de Sarasa's text, will be able to tell me
whether the modern writers I have cited have been reading too much into de
Sarasa's use of the term "logarithm", or whether the earlier writers have
failed to give credit where it was due.
Dhombres, in Mathematique au fil des Ages, page 188, tantalisingly quotes
de Sarasa as writing "the areas are like logarithms"; which leaves the
interpretation open!
Any help with finding sources, or with other clarifying information will be
much appreciated.
Bob Burn
Robert P. Burn
Professor of Mathematics Education
Dept of Mathematics
Agder College
Tordenskjoldsgate 65
4604 Kristiansand
Norway
Tel.(+47)-38-14-16-31
Fax.(+47)-38-14-10-71
email Robert.P.Burn@hia.no