Abstract
In a July 1798 entry in his mathematical diary, Gauss announced:
On the lemniscate, we have found out the most elegant things exceeding all expectations and that by methods which open up to us a whole new field ahead.
Paving the way to the new field of elliptic integrals predicted by Gauss was an elegant relationship that he discovered between three particular numerical values: the ratio of the circumference of a circle to its diameter (π), a particular value of an elliptic integral associated with the lemniscate \(\left (\int _0^1 \frac {1}{\sqrt {1-t^4}} dt \right )\), and the arithmetic-geometric mean of 1 and \(\sqrt {2}\). As an example of the powerful role which analogy and numerical experimentation can play within mathematics, the tale of Gauss’ path to these discoveries is one well worth sharing with today’s students. This paper describes a set of three “mini-Primary Source Projects” based on excerpts from Gauss’ mathematical diary and related manuscripts which are designed to tell that tale, while also serving to consolidate student proficiency with several standard topics studied in first-year calculus courses.
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Notes
- 1.
A fourth mini-PSP entitled Gaussian Guesswork: Arc Length and the Numerical Approximation of Integrals is planned for later development, with a projected completion date of Summer 2020.
- 2.
- 3.
The English translation of the paper’s full title is “Construction of a Curve with Equal Approach and Retreat, with the help of the rectification of a certain algebraic curve: Addendum to the June solution.”
- 4.
The paracentric isochrone is the curve with the property that a ball rolling down it approaches or recedes from a given point with uniform velocity. Although its construction did not require the finding of an area, it was viewed as a quadrature problem because the differential equation given by the physics involved led to the evaluation of integral (as would a quadrature problem). The paracentric isochrone problem itself was originally posed by Gottfried Leibniz in 1689 as one of a series of problems involving balls rolling along curves with which geometers of the day challenged each other, as well as the power of the new calculus techniques. In addition to the two solutions published by Jacob Bernoulli in June and September of 1694, Johann Bernoulli (1667–1748) also independently solved it in that same year by rectifying the lemniscate. This incident of independent discovery, in which Jacob beat Johann to publication by just a month, along with the surrounding debate about proper methodology in which the two brothers participated on different sides of the argument, were factors in the increasingly uncivil sibling rivalry that existed between the two brothers.
- 5.
Bernoulli’s full view about the best method to use was more complicated than we have presented here, as was the debate in general among geometers of the time. For a thorough treatment of these issues, see (Blåsjö 2017, 160–167).
- 6.
All excerpts from Gauss’ diary are taken from the English translation by Jeremy Grey that appears as an appendix in (Dunnington 2004, pp. 469–484).
- 7.
This is the curve that Bernoulli called the “elastica.”
- 8.
The symbol Gauss used here is a variant of the Greek letter π which is called “varpi.”
- 9.
Gauss reminisced about his 1791 discovery of this idea in a letter (Gauss 1816) that he wrote to his friend Schumacher much later, in 1816. Although his memory of the exact date may not be accurate, Gauss was certainly familiar with the arithmetic-geometric mean by the time he began his mathematical diary in 1796.
- 10.
The algorithm for computing the arithmetic-geometric mean did appear in a 1785 paper by Lagrange on the calculation of integrals of the form \(\displaystyle {\int \frac {Mdy}{\sqrt {(1+p^2y^2)(1+q^2y^2)}}}\) (Lagrange 1785). However, Gauss was unaware of Lagrange’s earlier independent discovery, and Lagrange himself did not appear to realize the full potential of the arithmetic-geometric mean.
- 11.
Gauss himself used prime notation (i.e., a ′, a ′′, a ′′′ ) to denote the terms of the sequence. In the project (and this paper), we instead use indexed notation (i.e., a 1, a 2, a 3 ) in keeping with current notational conventions. To fully adapt Gauss’ notation to that used today, we could also write a 0 = a and b 0 = b.
- 12.
Gauss’ use of superscripts (a ∞, b ∞) to denote the limiting values is again replaced with subscripts (a ∞, b ∞) throughout the project (and this paper).
- 13.
Gauss approximated the value of \(\varpi = 2\int _0^1 \frac {1}{\sqrt {1-t^4}} dt \) using power series methods. These techniques will be the focus of the fourth Gaussian Guesswork mini-PSP described in footnote 1.
- 14.
The terminology “elliptic integral” is introduced in all three mini-PSPs as the special cases of n = 3, n = 4 for integrals of the form \(\int _0^x \frac {1}{\sqrt {1-t^n}} dt\). For n ≥ 5, this integral is called hyperelliptic.
- 15.
Gauss’ use of prime notation (i.e., m ′, m ′′, m ′′′ ) to denote the terms of these sequences is again replaced here by indexed notation (i.e., m 1, m 2, m 3 ).
- 16.
Gauss’ use of degree notation for the limits of integration has been replaced by radians throughout the project (and this paper), in order to reduce the potential for unnecessary confusion on the part of students and instructors.
- 17.
If you enjoy algebraic challenges, try your hand at finding your own path for carrying out this substitution before reading further!
- 18.
The arithmetic-geometric mean is also used today to construct fast algorithms for calculating values of elementary transcendental functions and some classical constants, like π.
- 19.
For details about Gauss’ work on the arithmetic-geometric mean within the complex domain, see (Cox 1984). Cox also treats Gauss’ early work on the real-valued case, and explores some of the pre-Gaussian history of these ideas.
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Acknowledgements
The development of this paper has been partially supported by the National Science Foundation’s Improving Undergraduate STEM Education Program under Grant No. 1523494. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation. The author wishes to thank George W. Heine III for technical assistance with the Latin-to-English translations of the primary source excerpts that appear in this paper and with the reproduction of the images that appear in Figs. 1 and 2. She is also immensely grateful to Adrian Rice for the inspiration that his Math Horizons paper (Rice 2009) provided.
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Barnett, J.H. (2020). A Gaussian Tale for the Classroom: Lemniscates, Arithmetic-Geometric Means, and More. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-31298-5_9
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