Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the other two main branches, "algebra" and "geometry", which do not seem to have other unrelated meanings.

  • $\begingroup$ Algebra in a broad sense is the study of system of symbols and rules for manipulating these symbols. In what world is this not a generic term? $\endgroup$ – David H May 9 '14 at 6:01
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    $\begingroup$ I guess the OP means that analysis literally just means resolving, a very generic concept. I recall a similar question here regarding calculus; there the confusion was caused by the fact that this is just short for differential calculus. I don't think anything like that (e.g. differential analysis) was ever in use, but see the distinction from functional analysis and also analysis situs (today called topology) $\endgroup$ – Hagen von Eitzen May 9 '14 at 6:05
  • $\begingroup$ I always thought it took its name from analytic functions. $\endgroup$ – Jonathan Hebert May 9 '14 at 8:36
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    $\begingroup$ @Doop, the term "analysis" was being used in this sense before the notion of "function" was introduced, and certainly long before the notion of "analytic function" was introduced. $\endgroup$ – Mikhail Katz May 9 '14 at 9:03
  • $\begingroup$ I've never heard the term analysis being used by itself as a topic like that, only phrases like Linear Analysis. $\endgroup$ – starsplusplus May 9 '14 at 12:22

There is a tradition on early modern mathematics regarding the usage of the term analysis :

  • François Viète, Isagoge in Artem Analyticem (Introduction to the Analytic Art), Tours, 1591 (several successive editions and translations);

  • Thomas Harriot, Artis Analyticae Praxis, London, 1631.

The background is the "rediscovery" of ancient Greek mathematics and, in particular of Pappus of Alexandria, (c.A.D. 290 – c.350) and his main work in eight books titled Synagoge or Collection, which Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem.

See Henk Bos, Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction (2001), page :

Two kinds of analysis were distinguished in early modern geometry: the classical and the algebraic. The former method was known from examples in classical mathematical texts in which the constructions of problems were preceded by an argument referred to as "analysis;" in those cases the constructions were called "synthesis".

Reference to Pappus' problems is also found into René Descartes' La Géométrie (1637).

The two main line to be understood are :

  • analysis as a "method" to solve problem

  • analysis as the technique of treating geometrical problems with algebraic methods.

Both, I think, are "involved" into the use of analysis to name the new method introduced by Newton and Leibniz.

You can see :

Jaakko Hintikka & U.Remes, The Method of Analysis: Its Geometrical Origin and Its General Significance (1974)

and :

Michael Otte & Marco Panza, Analysis and Synthesis in Mathematics: History and Philosophy (1997).

  • $\begingroup$ Very nice references. It looks like you thought of the analysis/synthesis distinction a bit. If you are familiar with Fermat's two derivations of the least time principle in optics, I would appreciate hearing why you think he called them respectively "analysis" and "synthesis" as I only have a vague understanding of this. $\endgroup$ – Mikhail Katz May 9 '14 at 8:51
  • $\begingroup$ THe issue is "complicated" because dates back to Greek math (I'm not very familiar with it) and the Renaissance discovery of (the heritage of) Greek math. One of the "threads" was the (partly misconceived) convinction that there were hidden in some Greek book a "method" (sse Archimedes) for the discovery of the solution of math problems: this was the analytical side, compared to the exposition of the theorem in axiomatic form typical of Greek masterpieces (Euclid's Elements): this was the Synthetic side. During Renaissance there were a lot of debate both in philosophy and math. 1/2 $\endgroup$ – Mauro ALLEGRANZA May 9 '14 at 9:07
  • $\begingroup$ I think can be useful Bos's book, Paolo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (1999) and Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1932 - Dover reprint). 2/3 $\endgroup$ – Mauro ALLEGRANZA May 9 '14 at 9:11
  • $\begingroup$ Descartes is (for me) a "big player" in this issue, both for his mathematical innovation (that must be read as a method for the "discovery" of the solution of geometrical problems) and for his philosophical backround regarding method (see the unpublished Regulae ad directionem ingenii (1626–1628): see Rule five). I do not know about Fermat's optics, but I think this background may help. 3/3 $\endgroup$ – Mauro ALLEGRANZA May 9 '14 at 9:21
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    $\begingroup$ @user72694 - I'mm alluding to your post regarding l'Hopital; my refernce is that in order to understand Fermat's double proof, we have to study the context. Half a century before the first "systhematization" (l'Hopital textbook) there were confidence about the power of "infinitesimal" method, but at the same time difficulties in justifying its soundness; thus the usage (by Fermat and others) as a tool for discovery the solution of a problem (analysis) to be supplemented by a "regular" proof of it (synthesis). $\endgroup$ – Mauro ALLEGRANZA May 11 '14 at 9:22

The first occurrence (1696) of the term "analysis" in the sense of the mathematical discipline extending calculus occurs in the title of l'Hopital's work Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes. The term had been used earlier as part of a dichotomy analysis/synthesis, for example in Fermat. However l'Hopital was the first to use the term to describe the new science being created by Leibniz and others in the 17th century.


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