I have begun to examine thoroughly the ~~elastic~~ lemniscatic^{1} curve depending on \(\int \left (1 – x^4 \right)^{-1/2} dx\).

Gauss’ *Mathematical **Diary,*^{2} January 8, 1797

At first glance, the integral in this diary entry will look to many students like just another of the possibly hundreds of integrals they encountered in Calculus 2—a course that can seem filled by hours of mundane practice with a dizzying array of techniques and concepts that leave little room for imagination or invention. Hidden from view are the origins of those techniques and concepts in the inventive imaginations of mathematicians such as Carl Friedrich Gauss (1777–1855), for whom the examination of the integral *\(\int \left (1 – x^4 \right)^{-1/2} dx\) *led to the development of deeply profound and beautiful mathematical results. The three mini-Primary Source Projects (PSPs) presented in this article offer a glimpse of those results within the context of standard techniques and concepts from Calculus 2. Entitled *Gaussian Guesswork*—and inspired by the article of the same title by Adrian Rice [2009]—each of these three mini-PSPs further offer students an opportunity to witness and experience how experimentation, observation and analogy can play a role in mathematical practice.

Diagram reproduced from [Bernoulli 1695] showing the lemniscate with the paracentric isochrone.

The curve that Gauss originally called the elastic curve was christened with the name *lemniscate* by Jacob Bernoulli (1655–1705), in connection with his construction of another curve called the paracentric isochrone. That construction relied on the arclength of the lemniscate, which in turn relies on the very integral that interested Gauss in 1797. In the mini-PSP *Gaussian Guesswork: Polar Coordinates, Arc Length and the Lemniscate Curve*, students begin by examining the less difficult integral associated with arclengths within the unit circle. Two particular facts about that arclength integral are highlighted:

*\(\int_0^1 \frac{1}{\sqrt{1 – x^2}} dx\)* gives the arclength of one-fourth of the unit circle; that is, *\(\int_0^1 \frac{1}{\sqrt{1 – x^4}} dx =\frac{\pi}{2}\)*; and
- the familiar sine function is the inverse of the function defined by
*\(f(t) = \int_0^t \frac{1}{\sqrt{1 – x^2}} dx\)*.

After verifying these details through a few preliminary tasks, students are led (again, by way of project tasks) through the process of using polar coordinates to show that the integral for the arclength of one quarter of the unit lemniscate is given by *\(\int_0^1 \frac{1}{\sqrt{1 – x^4}} dx\)*. The project then turns to a brief survey of the conclusions that Gauss derived from the analogy between the two arclength integrals in his paper [Gauss 1797a]; these included:

- the introduction of a new quantity, denoted \(\varpi\) and defined as the arclength of one-fourth of the unit lemniscate; that is,
*\(\int_0^1 \frac{1}{\sqrt{1 – x^4}} dx=\frac{\varpi}{2}\)*; and
- the definition of the
*lemniscatic sine* function as the inverse function *\(g(t) = \int_0^t \frac{1}{\sqrt{1 – x^4}} dx\).*

Sketch of the lemniscate from Gauss’ *Nachlass* [Gauss 1797b, 160].

The numbers \(\pi\) and \(\varpi\) that feature in the mini-PSP described above are two of the three quantities that played a role in Gauss’ guesswork related to the integral* \(\int \left (1 – x^4 \right)^{-1/2} dx\)*. He announced the third quantity in this numerical trio in a later diary entry:

We have established that *the arithmetic-geometric mean between \(\sqrt{2}\) and 1* is *\(\pi/\varpi\) *to 11 places;

the proof of this fact will certainly open up a new field of analysis.

Gauss’ *Mathematical Diary, *May 30, 1799 (emphasis added)

Gauss appears to have discovered the arithmetic-geometric mean when he was only 14 years old,^{3} but he published very little about it during his lifetime. The mini-PSP *Gaussian Guesswork: **Infinite Sequences and the Arithmetic-Geometric Mean *draws on a paper that was published posthumously as part of his *Nachlass** *[Gauss 1799]. Students begin this project by reading Gauss’ definition of the two sequences needed to define the arithmetic-geometric mean:^{4}

\[\left \{\begin{array}{cccccccc}a_0, &a_1,&a_2 &a_3, \ldots\\b_0, &b_1,&b_2, &b_3, \ldots\end{array}\right\}\]

. . . [where] the terms of the upper sequence have the value of the arithmetic mean, and those of the lower sequence, the geometric mean:

\[a_1=\frac{1}{2}(a+b), \,\,b_1=\sqrt{ab}, \,\, a_2=\frac{1}{2}(a_1+b_1), \,\,b_2=\sqrt{a_1b_1}, \,\, a_3=\frac{1}{2}(a_2+b_2), \,\,b_3=\sqrt{a_2b_2}. \,\, \]

After working through his first two numerical examples of such sequences and making some observations of their own, students read Gauss’ observations about those same examples and provide proofs of the general properties that he derived from his observations. Here, the Monotone Convergence Theorem arises organically as the natural means to prove the sequences \((a_n)\) and \((b_n)\) are both convergent. The proof that their limits are in fact equal then naturally leads to the definition of the *arithmetic-geometric mean between the initial values \(a\), \(b\)* as that common limit value.

Gauss’ final example in his *Nachlass *paper [Gauss 1799] brings in the third number of his numerical trio: the arithmetic-geometric mean between \(\sqrt{2}\) and \(1\).

Example 4: \( a = \sqrt{2}\), \(b = 1\) \[\begin{array}{lll} a_0 \,=\, 1.41421\,35623\,73095\,04880\,2 & \hspace{10pt} & b_0 \, = \, 1.00000\,00000\,00000\,00000\,0 \\ a_1 = 1.20710\,67811\,86547\,52440\,1 & \hspace{10pt} & b_1 = 1.18920\,71150\,02721\,06671\,7\\ a_2 =1.19815\,69480\,94634\,29555\,9 & \hspace{10pt} & b_2 =1.19812\,35214\,93120\,12260\,7\\ a_3= 1.19814\,02347\,93877\,20908\,3 & \hspace{10pt} & b_3 = 1.19814\,02346\,77307\,20579\,8\\ a_4 = 1.19814\,02347\,35592\,20744\,1 & \hspace{10pt} & b_4= 1.19814\,02347\,35592\,20743\,9 \end{array}\]

Here, we see also the reason behind the full title of Rice's article [Rice 2009]:

“Gaussian Guesswork, or why 1.19814023473559220744… is such a beautiful number.”

The mini-PSP *Gaussian Guesswork: Elliptic Integrals and Integration by Substitution* reveals another connection to this beautiful number by taking up the difficult question of how to evaluate the integral *\(\int_0^1 \left (1 – x^4 \right)^{-1/2} dx\) *that first sparked Gauss' interest in 1796. The rather surprising result? Letting \(\mu\) denote the arithmetic-geometric mean between \(\sqrt{2}\) and \(1\),

*\[\int_0^1 \left (1 – x^4 \right)^{-1/2} dx = \frac{1}{\mu}.\]*

Gauss derived this result from a more general integration theorem that he proved in an important astronomical paper on the gravitational attraction of planets [Gauss 1818]. Although the sophisticated substitution that he used in that proof is not itself part of the standard Calculus 2 curriculum, working through its details provides an excellent opportunity for beginning calculus students to apply and consolidate core concepts and techniques while witnessing their interplay within the context of some amazingly beautiful, surprising and important mathematics. Indeed, what began as an examination of a single integral combined with some Gaussian guesswork about a numerical relationship between three numbers not only opened up the field of elliptic functions of a single real-valued variable, but also led Gauss well into the realm of functions of several complex-valued variables and beyond.

Excerpt from facsimile copy of Gauss’ original diary, from Volume X.1 of Gauss’ *Werke*.

All three *Gaussian Guesswork* mini-PSPs are ready for student use:

Each tells a portion of the tale in some detail, and provides enough of a glimpse of the remaining ideas to give students a feel for the complete story. Any of the three can thus be used either alone or in conjunction with either or both of the others. A set of instructor notes offering practical advice for classroom use is appended at the end of each project. The LaTeX source code of each project is also available from the author by request.

These projects mark the twenty-first entry in *A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources *appearing in *Convergence*, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in this series appear below. The full TRIUMPHS collection also offers 10 other mini-PSPs for use in teaching courses in the first-year calculus sequence.

**Notes**

[1] Originally, Gauss wrote ‘elastic’ here, only to cross it out at some later unknown date when he instead wrote in ‘lemniscatic.'

[2] Gauss’ diary remained in the possession of his family until 1898 and was first published by Felix Klein (1849–1925) in [Klein 1903]. An English translation with commentary on its mathematical contents by the historian of mathematics Jeremy Gray appears in [Gray 1984], and was later reprinted in [Dunnington 2004, 469–496]. A facsimile of the original diary can also be found in Volume X.1 of Gauss’ *Werke* (Collected Works).

[3] Gauss himself reminisced about his 1791 discovery of this idea in a letter, [Gauss 1816], that he wrote to his friend Heinrich Christian Schumacher (1780–1850) much later. Although his memory of the exact date of his discovery may not have been accurate when he wrote that 1816 letter, Gauss was certainly familiar with the arithmetic-geometric mean by the time he began his mathematical diary in 1796.

[4] Gauss himself used prime notation (i.e., \(a'\), \(a''\), \(a'''\), \(a''''\)) to denote the terms of the sequence. In the project, indexed notation (i.e., \(a_1\), \(a_2\), \(a_3\), \(a_4\)) in keeping with current notational conventions is used instead.

**Acknowledgements**

The author is grateful to Adrian Rice, whose *Math Horizons *article “Gaussian Guesswork, or why 1*.*19814023473559220744 …* *is such a beautiful number” [Rice 2009] inspired the development of these projects.

The development of the student projects presented in this article has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from 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.

**References**

Bernoulli, Jacob. 1695, December. Explicationes, annotationes et additiones ad ea quæ in Actis superiorum annorum de Curva Elastica, Isochrona Paracentrica, & Velaria, hin inde memorata, & partim controversa lenuntur; ubi de Linea mediarum directionum, aliisque novis (Explanations, notes and additions to that in the Acts of the preceding year about the Elastic, Paracentric Isochrone and Velara Curves, thence from this recounted, the controversial part read, where concerning the line of the middle directions). *Acta Eruditorum*: 537–553. Also in G. Cramer, editor, *Opera Omnia*, Volume 1, pages 639–662. Geneva: Cramer, 1744.

Dunnington, Guy Waldo. 2004. *Carl Friedrich Gauss: Titan Of Science*. Washington, DC: The Mathematical Association of America. Reprint of original 1955 publication. Includes the English translation of Gauss’ *Mathematical Diary* by Jeremy Gray.

Gauss, Carl Friedrich. 1797a. Elegantiores Integralis \(\int_0^1 \frac{dx}{\sqrt{1 – x^4 }}\) Proprietates (Very Excellent Properties of the Integral \(\int_0^1 \frac{dx}{\sqrt{1 – x^4 }}\)). In Ernst Schering, editor, *Werke*, volume III, pages 404–412. Göttingen: Gedruckt in der Dieterichschen universitätsdruckerei, 1866.

Gauss, Carl Friedrich. 1797b. Teilung der Lemniskate (Division of the Lemniscate). In Felix Klein, editor, *Werke*, volume X.1, pages 160–164. Göttingen: Gedruckt in der Dieterichschen universitätsdruckerei, 1917.

Gauss, Carl Friedrich. 1799. Arithmetisch Geometrisches Mittel (Arithmetic-Geometric Mean). In Ernst Schering, editor, *Werke*, volume III, pages 361–432. Göttingen: Konigliche Gesellschaft der Wissenschaft, 1866.

Gauss, Carl Friedrich. 1816. Letter from Gauss to Schumacher dated April 1816 (in German). In Felix Klein, editor,* Werke*, volume X.1, pages 247–248. Göttingen: Konigliche Gesellschaft der Wissenschaft, 1917.

Gauss, Carl Friedrich. 1818. Determinatio attractionis, quam in punctum quodvis positionis datae exerceret planeta, si eius massa per totam orbitam ratione temporis, quo singulae partes descibuntur, uniformiter esset dispertita (Determination of the Attraction, which a planet exerts on any point, if its mass is distributed uniformly through the time of the orbit). Presented to the *Göttingen Royal Society of Science *(January 17, 1818). Also in Ernst Schering, editor, *Werke*, volume III, pages 331–356. Göttingen: Gedruckt in der Dieterichschen universitätsdruckerei, 1866.

Gray, J. J. 1984. A Commentary on Gauss’s Mathematical Diary, 1796–1814, with an English Translation. *Expositiones Mathematicae* 2(2):97–130.

Klein, Felix. 1903. Gauss’ Wissenschaftliches Tagebuch (Mathematical Diary), 1796–1814. *Mathematische Annalen* 57:1–34.

Rice, Adrian. 2009, November. Gaussian Guesswork, or why 1.19814023473559220744 . . . is such a beautiful number. *Math Horizons*: 12–15.