One striking feature of nineteenth century mathematics, as contrasted with that of previous eras, is...

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One striking feature of nineteenth century mathematics, as contrasted with that of previous eras, is the higher degree of rigor and precision demanded by its practitioners. This tendency was especially noticeable in analysis, a field of mathematics that essentially began with the “invention” of calculus by Leibniz and Newton in the mid-17th century. Unlike the calculus studied in an undergraduate course today, however, the calculus of Newton, Leibniz and their immediate followers focused entirely on the study of geometric curves, using algebra (or ‘analysis’) as an aid in their work. This situation changed dramatically in the 18th century when the focus of calculus shifted instead to the study of functions, a change due largely to influence of the Swiss mathematician and physicist Leonhard Euler (1707–1783). In the hands of Euler and his contemporaries, functions became a powerful problem solving and modelling tool in physics, astronomy, related mathematical fields such as differential equations and the calculus of variations. Why then, after nearly 200 years of success in the development and application of calculus techniques, did 19th-century mathematicians feel the need to bring a more critical perspective to the study of calculus? This project explores this question through selected excerpts from the writings of the 19th century mathematicians who led the initiative to raise the level of rigor in the field of analysis.
	
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Nineteenth Century Mathematics and the Origins of Analysis

Janet Heine Barnett
December 22, 2016
One striking feature of nineteenth century mathematics, as contrasted with that of previous
eras, is the higher degree of rigor and precision demanded by its practitioners. This tendency was
especially noticeable in analysis, a eld of mathematics that essentially began with the \invention"
of calculus by Leibniz and Newton in the mid-17th century. Unlike the calculus studied in an
undergraduate course today, however, the calculus of Newton, Leibniz and their immediate followers
focusedentirelyonthestudyofgeometric curves,usingalgebra(or`analysis')asanaidintheirwork.
This situation changed dramatically in the 18th century when the focus of calculus shifted instead
to the study of functions, a change due largely to inuence of the Swiss mathematician and physicist
Leonhard Euler (1707{1783). In the hands of Euler and his contemporaries, functions became a
powerful problem solving and modelling tool in physics, astronomy, related mathematical elds such
as di?erential equations and the calculus of variations. Why then, after nearly 200 years of success
in the development and application of calculus techniques, did 19th-century mathematicians feel
the need to bring a more critical perspective to the study of calculus? This project explores this
question through selected excerpts from the writings of the 19th century mathematicians who led
the initiative to raise the level of rigor in the eld of analysis.
1 The Problem with Analysis: Bolzano, Cauchy and Dedekind
To begin to get a feel for what mathematicians felt was wrong with the state of analysis at the
start of the 19th century, we will read excerpts from three well-known analysts of the time: Bernard
Bolzano (1781{1848), Augustin Cauchy (1789{1857) and Richard Dedekind (1831{1916). In these
excerpts, these mathematicians expressed their concerns about the relation of calculus (analysis)...

Robert · Accepted Answer

1. Bernard Bolzano is clearly against the use of 
geometry in proving theorems in calculus. 
Augustin Cauchy was not a supporter of the 
method of Algebra that was used to prove 
theorems in calculus. 
Richard Dedekind considered the method of 
geometry useful but not scientific and hence 
wanted to prove theorems in calculus rigorously 
using arithmetic.
2. (a).Bolzano first states that between any two 
quantities with opposite sign there is always a root 
of the equation. 
Next he stated that every algebraic rational 
integral of one variable quantity can be divided 
into real factors of first or second degree.
. Richard Dedekind stated that every magnitude which 
grows continually but not beyond all limits has to 
approach a limiting value. 
. Augustin Cauchy rejected divergent series expansions 
and then he deferred Taylor’s formula to an integral. 
Also he stated that in some of the cases where Taylor’s 
series converges its sum differs from the function 
concerned.

One striking feature of nineteenth century mathematics, as contrasted with that of previous eras, is the higher degree of rigor and precision demanded by its practitioners. This tendency was...

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