Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Annotations to a certain passage of Descartes for finding the quadrature
  of the circle   Translation from the Latin of "Annotationes in locum quendam Cartesii ad
circuli quadraturam spectantem" (1763).
  The passage Euler is referring to is the "Excerpta" in part 6, p. 6 of
Descartes' 1701 "Opuscula posthuma".
  Before reading this paper I had not heard of the "quadratrix" before, and I
recommend learning a bit about it before reading this. I found Thomas Heath, "A
history of Greek mathematics", vol. I, chapter VII to be helpful, in particular
pp. 226-230. The quadratrix is a "mechanical curve" that can be used to rectify
the circle. The usual problem of squaring the circle is to construct a square
with the same area (or perimeter) as a given circle, in a finite number of
steps using compass and straightedge. Descartes worked in the reverse
direction: from a given square he constructed the radius of a circle with the
same perimeter, but in an infinite number of steps.
  In this paper Euler reconstructs Descartes' argument and develops some
consequences of it. Euler finds that \[ \sum_{n=0}^\infty \frac{1}{2^n} \tan
\frac{1}{2^n}\phi = \frac{1}{\phi} - 2\cot 2\phi. \] Integrating this yields \[
\prod_{n=1}^\infty \sec \frac{1}{2^n} \phi = \frac{2\phi}{\sin 2\phi}. \]
  I'd like to thank Davide Crippa from the University of Paris 7 for some
helpful back and forth about this paper.
  One of the only citations to this paper that I have found is in Pietro
Ferroni, De calculo integralium exercitatio mathematica, Allegrini, Florence,
1792, pp. xxi--xxiii. The full text of it is available on Google Books.

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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
3
The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph