Carl Friedrich Gauss

How to use this course

Carl Friedrich Gauss worked for six decades and left a mark on almost every part of mathematics and much of physics. A full survey would fill a library. This course follows his main lines of work in roughly the order he took them up: number theory and algebra in his youth, astronomy and statistics in his thirties, geodesy and differential geometry in his forties, and physics in his fifties. Each strand is still alive in modern mathematics, and the lessons try to show the bridge from his formulation to today’s.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Gauss wrote in Latin and later in German, with his own notation. We use modern notation throughout and note where it differs from his.

Timeline

Brunswick, 1777–1795

Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, the capital of a small German duchy. His father, Gebhard Dietrich Gauss, worked at a series of manual trades: gardener, bricklayer, butcher, and keeper of accounts for a burial society. His mother, Dorothea Benze, could barely read. He had an elder half-brother from his father’s first marriage. Nothing in the household pointed toward learning.

The stories of his early ability are many and mostly come from Gauss himself in old age. The best known is set in his primary school, where the teacher J. G. Büttner set the class to add the integers from 1 to 100 and the young Gauss found the total at once by pairing terms. Whatever the exact details, Büttner and his young assistant Martin Bartels saw that the boy needed more than the school could give. Bartels, later a professor at Kazan where he taught Nikolai Lobachevsky, studied with him and helped him into the Gymnasium in 1788.

The Duke’s stipend

In 1791 Gauss was presented to Carl Wilhelm Ferdinand, Duke of Brunswick, who granted him a stipend. It paid for three years at the Collegium Carolinum (1792–1795), a school between a gymnasium and a university, and later for his studies at Göttingen. The Duke’s support lasted until his death in 1806, and Gauss dedicated the Disquisitiones Arithmeticae to him.

At the Collegium Gauss read Newton, Euler and Lagrange, and began a habit he kept for life: computing large tables by hand to find patterns. By counting primes in blocks he arrived, at about fifteen, at the guess that the density of primes near \(x\) is about \(1/\log x\), the content of the prime number theorem. He also rediscovered quadratic reciprocity empirically and began work on the arithmetic-geometric mean. None of this was yet proved.

Göttingen and the Disquisitiones, 1795–1801

In October 1795 Gauss went to the University of Göttingen, chosen over the Duke’s own university at Helmstedt for its library. The professor of mathematics was Abraham Gotthelf Kästner, whom Gauss did not admire. His closest friend there was the Hungarian student Farkas Bolyai, with whom he corresponded for the rest of his life.

He was still undecided between mathematics and classical philology. The question was settled on 30 March 1796, when he found that the regular 17-gon can be constructed with ruler and compass (Lesson 04). No new constructible polygon had been found since Euclid. On the same day he began a mathematical diary. Its 146 short entries, kept until 1814, record discoveries he often never published. One entry of July 1796 reads, in effect, “Eureka: every number is a sum of three triangular numbers.” The diary was found in 1897 and published by Felix Klein in 1903.

Degree and book

Gauss left Göttingen in 1798 without a degree. At the Duke’s wish he took a doctorate at Helmstedt, where Johann Friedrich Pfaff, the leading German mathematician of the day, assessed his thesis. It was awarded in 1799 without an oral examination. The thesis gave a proof of the fundamental theorem of algebra and a sharp critique of earlier attempts by d’Alembert, Euler and Lagrange (Lesson 07).

His great book, the Disquisitiones Arithmeticae, appeared in Leipzig in 1801 after delays at the printer. Its seven sections cover congruences, primitive roots, quadratic reciprocity with the first complete proof, the theory of binary quadratic forms, and the division of the circle. It turned number theory from a collection of results by Fermat, Euler, Lagrange and Legendre into a systematic subject. A planned eighth section was never published in his lifetime.

Ceres and the move to Göttingen, 1801–1809

On 1 January 1801 Giuseppe Piazzi at Palermo found a faint moving object, which he named Ceres. He followed it for a few weeks before it vanished into the glare of the Sun. Astronomers had only a short arc of observations and no method that could fit an orbit to them. Gauss, aged 24, developed a new method of orbit determination and sent a prediction to Franz Xaver von Zach. On 31 December 1801 von Zach found Ceres close to the predicted place, and Heinrich Olbers confirmed it the next evening.

The success made Gauss famous across Europe. Olbers became a close friend and a lifelong correspondent, and through astronomy Gauss came to know Friedrich Wilhelm Bessel, whose career he later helped by arranging an honorary doctorate from Göttingen in 1811. From 1804 he corresponded with a “Monsieur Le Blanc” on number theory. In 1807, when French troops occupied Brunswick, the writer revealed herself as Sophie Germain, who had asked a French general to make sure Gauss came to no harm.

Family and appointment

On 9 October 1805 Gauss married Johanna Osthoff, the daughter of a Brunswick tanner. Their son Joseph was born in 1806 and their daughter Wilhelmine in 1808. In 1806 Duke Carl Wilhelm Ferdinand was mortally wounded at Jena-Auerstedt, and Gauss’s stipend was at risk. He accepted an offer from Göttingen, then in the Kingdom of Westphalia, and in November 1807 became professor of astronomy and director of the observatory.

The year 1809 brought both triumph and disaster. He published Theoria motus corporum coelestium, the systematic account of his orbit methods, with the method of least squares and the normal law of errors (Lesson 08). On 11 October Johanna died, a month after giving birth to their son Louis, who died a few months later. In 1810 Gauss married her friend Minna Waldeck. They had three children: Eugen, Wilhelm and Therese.

The 17-gon and cyclotomy

For the general reader. The Greeks could construct regular polygons with 3, 4, 5 and 15 sides using ruler and compass, and could double the number of sides. For two thousand years nobody added to the list. At eighteen Gauss showed that a 17-sided polygon can also be drawn, and explained exactly which polygons can.

Constructibility

A point is constructible from \(0\) and \(1\) exactly when its coordinates lie in a tower \(\mathbb Q=K_0\subset K_1\subset\cdots\subset K_m\) with each \([K_{i+1}:K_i]=2\). Constructing the regular \(n\)-gon is the same as constructing \(\cos(2\pi/n)\), or the complex number \(\zeta_n=e^{2\pi i/n}\).

Gauss proved sufficiency in Section VII of the Disquisitiones and stated necessity; Pierre Wantzel published a proof of necessity in 1837. The known Fermat primes are 3, 5, 17, 257 and 65537.

Gauss periods for \(p=17\)

Let \(p\) be prime and \(\zeta=\zeta_p\). The powers \(\zeta,\zeta^2,\dots,\zeta^{p-1}\) are permuted cyclically by \(\zeta\mapsto\zeta^g\) for a primitive root \(g\) mod \(p\). For \(p=17\) take \(g=3\); the powers of 3 mod 17 are

Gauss summed \(\zeta^k\) over every second exponent in this list, giving two periods of eight terms:

The first set is the squares mod 17. Then \(\eta_0+\eta_1=-1\), and a direct count gives \(\eta_0\eta_1=-4\). So \(\eta_0,\eta_1\) are the roots of \(x^2+x-4=0\):

Splitting each period again into two periods of four terms, then of two terms, gives quadratic equations at each stage. The two-term period \(\zeta+\zeta^{-1}=2\cos(2\pi/17)\) is reached after four square roots. The final formula is

In modern terms, \(\mathrm{Gal}(\mathbb Q(\zeta_p)/\mathbb Q)\cong(\mathbb Z/p)^\times\) is cyclic of order \(p-1\), and the periods generate the fixed fields of its subgroups. When \(p-1=2^N\) there is a chain of subgroups of index 2, and so a tower of quadratic extensions. Gauss found this structure thirty years before Évariste Galois.

Congruences and quadratic reciprocity

For the general reader. Gauss introduced the notation \(a\equiv b \pmod m\), which treats remainders as numbers in their own right. With it he proved a law that Euler and Legendre had seen but could not establish: whether \(p\) is a perfect square in the arithmetic of remainders mod \(q\) is tied by a simple rule to whether \(q\) is a square mod \(p\). Gauss called it the theorema aureum, the golden theorem.

For an odd prime \(p\) and \(a\not\equiv0\), the Legendre symbol \(\left(\frac ap\right)\) is \(+1\) if \(a\) is a square mod \(p\) and \(-1\) otherwise. Euler’s criterion says \(\left(\frac ap\right)\equiv a^{(p-1)/2}\pmod p\), so the symbol is multiplicative in \(a\).

Gauss gave the first proof in the Disquisitiones, by a long induction, and published or left eight proofs in all. The proof below uses the sums he studied in 1801–1811.

Proof with Gauss sums

Let \(\zeta=e^{2\pi i/p}\) and \(g=\sum_{a=1}^{p-1}\left(\frac ap\right)\zeta^a\). A short computation gives \(g^2=p^*\), where \(p^*=(-1)^{(p-1)/2}p\). Work in the ring \(\mathbb Z[\zeta]\) modulo \(q\). On one side,

On the other side, the Frobenius rule \((x+y)^q\equiv x^q+y^q\) gives

Since \(g^2=\pm p\) is invertible mod \(q\), we may cancel \(g\): \(\left(\frac qp\right)=\left(\frac{p^*}q\right)=\left(\frac{-1}q\right)^{\frac{p-1}2}\left(\frac pq\right)\), which is the law.

Quadratic forms and Gaussian integers

For the general reader. Which numbers can be written as \(x^2+y^2\), or \(x^2+5y^2\)? Fermat, Euler and Lagrange had partial answers. In the longest section of the Disquisitiones Gauss built a full theory of such expressions. Later he extended arithmetic itself to numbers \(a+bi\), and found that primes and factorization still make sense there.

Forms and reduction

A binary quadratic form is \(f(x,y)=ax^2+bxy+cy^2\) with integer coefficients and discriminant \(D=b^2-4ac\). (Gauss wrote \(ax^2+2bxy+cy^2\) and used the determinant \(b^2-ac\).) Two forms are properly equivalent if one becomes the other under a substitution \((x,y)\mapsto(\alpha x+\beta y,\gamma x+\delta y)\) with \(\alpha\delta-\beta\gamma=1\). Equivalent forms represent the same integers.

Gauss’s deepest contribution here is composition: a rule that combines two classes of discriminant \(D\) into a third. In modern language, the classes of primitive forms of discriminant \(D\) form a finite abelian group, isomorphic to the ideal class group of the quadratic order of discriminant \(D\). Gauss also grouped classes into genera according to the values they take modulo \(D\), and proved that the number of genera is a power of 2.

Gaussian integers

In his second memoir on biquadratic residues (1832) Gauss studied the ring \(\mathbb Z[i]=\{a+bi\}\), arguing that the law of biquadratic reciprocity can only be stated naturally there. He showed it has division with remainder for the norm \(N(a+bi)=a^2+b^2\), hence unique factorization.

Corollary (Fermat): an odd prime is a sum of two squares iff \(p\equiv1\pmod4\). Gauss’s picture of \(a+bi\) as a point in the plane, defended in his 1831 announcement of this memoir, did much to make complex numbers respectable.

The fundamental theorem of algebra and the AGM

For the general reader. Every polynomial equation has a solution among the complex numbers. Gauss’s doctoral thesis gave the first proof that most mathematicians accepted. In the same years he found a fast-converging process, the arithmetic-geometric mean, that led him to the theory of elliptic functions.

Gauss’s 1799 proof considered the real curves \(\operatorname{Re}p=0\) and \(\operatorname{Im}p=0\) and argued that they must cross. It relied on a topological fact about algebraic curves that he took as evident and that was fully justified only by Alexander Ostrowski in 1920. Gauss gave a purely algebraic second proof and an analytic third proof in 1816, and a fourth, for complex coefficients, at his jubilee in 1849.

A winding-number proof

Let \(p(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_0\) with no roots. For each \(r\ge0\) the loop \(\gamma_r(t)=p(re^{it})\), \(0\le t\le2\pi\), avoids 0. Its winding number about 0 is an integer depending continuously on \(r\), hence constant. For \(r=0\) the loop is constant and the winding number is 0. For large \(r\), \(|p(z)-z^n|<|z^n|\) on \(|z|=r\), so \(p\) winds about 0 as often as \(z^n\), namely \(n\) times. So \(n=0\).

The arithmetic-geometric mean

For \(a,b>0\) set \(a_0=a\), \(b_0=b\), \(a_{k+1}=\tfrac12(a_k+b_k)\), \(b_{k+1}=\sqrt{a_kb_k}\). Both sequences converge to a common limit \(M(a,b)\).

Proof idea. Call the integral \(I(a,b)\). Gauss showed by the substitution \(\sin\theta=\frac{2a\sin\phi}{a+b+(a-b)\sin^2\phi}\) that \(I(a,b)=I\big(\tfrac{a+b}2,\sqrt{ab}\big)\). Iterating, \(I(a,b)=I(a_k,b_k)\to I(M,M)=\frac{\pi}{2M}\). In 1799 he compared \(M(1,\sqrt2)\) with \(\pi/\varpi\) numerically to many decimal places and wrote in his diary that this opened an entirely new field of analysis. That field, the inversion of elliptic integrals into doubly periodic functions, he developed privately; Niels Henrik Abel and Carl Gustav Jacobi published it independently in the 1820s.

Least squares and the normal law

For the general reader. Every measurement has errors. When there are more measurements than unknowns, they cannot all be satisfied exactly. Gauss proposed choosing the answer that makes the sum of squared errors as small as possible, and justified it by a theory of how errors are distributed. The bell-shaped curve that came out of his argument still carries his name.

The method

Suppose \(y=Ax+\varepsilon\), with \(A\) an \(m\times n\) matrix of full rank \(n\le m\). The least-squares estimate minimizes \(\|y-Ax\|^2\). Setting the gradient to zero gives the normal equations

Geometrically, \(A\hat x\) is the orthogonal projection of \(y\) onto the column space of \(A\). Gauss solved such systems by the elimination now named after him.

Adrien-Marie Legendre published the method first, in 1805. Gauss stated in Theoria motus (1809) that he had used it since 1795, which Legendre resented. Gauss’s use in the Ceres computations makes his claim plausible, but the printed priority is Legendre’s.

Why squares: the 1809 argument

Gauss assumed that errors have a smooth density \(\varphi\) and that, for repeated measurements of a single quantity, the most probable value is the arithmetic mean. Maximizing \(\prod_i\varphi(y_i-x)\) at \(x=\bar y\) for all samples forces \(\varphi'(\Delta)/\varphi(\Delta)=-k\Delta\), so

and maximizing the likelihood then means minimizing \(\sum\Delta_i^2\). Pierre-Simon Laplace soon supplied a different justification through the central limit theorem.

This second justification, in Theoria combinationis observationum, was the one Gauss himself preferred.

Curved surfaces and the Theorema Egregium

For the general reader. An ant living on a surface, unable to look outside it, can still tell whether the surface is curved by measuring distances and angles. Gauss proved this in 1827 and called it the Theorema Egregium, the remarkable theorem. It explains why no flat map of the Earth can be accurate and it opened the road to Riemann’s geometry and to general relativity.

Let \(S\subset\mathbb R^3\) be a smooth surface with unit normal \(N\). Gauss defined curvature by the Gauss map \(N:S\to S^2\): the curvature at a point is the limiting ratio of the area of \(N(R)\) on the unit sphere to the area of a small region \(R\). Equivalently \(K=\kappa_1\kappa_2\), the product of the principal curvatures, or in terms of the two fundamental forms,

Gauss proved the theorem by direct calculation: the numerator \(LN-M^2\) can be rewritten, using the equations of the surface, in terms of \(E,F,G\) and their first and second derivatives alone. In geodesic polar coordinates \(ds^2=dr^2+G(r,\theta)\,d\theta^2\) the formula becomes \(K=-\frac{(\sqrt G)_{rr}}{\sqrt G}\).

Pierre Ossian Bonnet extended this in 1848 to regions with curved boundaries, and the global form \(\iint_S K\,dA=2\pi\chi(S)\) for closed surfaces is the Gauss–Bonnet theorem. Gauss’s work came out of his survey of Hanover (Lesson 11), where he had to project measurements on the curved Earth onto flat maps. He also thought deeply about the parallel postulate and convinced himself, without publishing, that a consistent non-Euclidean geometry exists.

Potential theory and magnetism

For the general reader. From 1831 Gauss worked with the physicist Wilhelm Weber on electricity and the Earth’s magnetism. He showed how to measure magnetic strength in absolute units of length, mass and time, and he wrote down general theorems about forces that fall off as the square of the distance. One of them, Gauss’s law, is one of Maxwell’s four equations.

Flux through a closed surface

Gauss had used the divergence theorem in special cases in his 1813 memoir on the attraction of ellipsoids. For a point mass or charge the key fact is the solid-angle computation.

Gauss’s 1840 memoir Allgemeine Lehrsätze developed the theory of the potential \(V\) with \(\mathbf E=-\nabla V\). Among its results: a harmonic function equals its average over any sphere in its domain, and the equilibrium distribution of charge on a conductor can be found by minimizing an energy integral, an early form of what Riemann later called the Dirichlet principle.

The geomagnetic field

In Allgemeine Theorie des Erdmagnetismus (1839) Gauss wrote the magnetic potential outside a sphere of radius \(a\) as a series of spherical harmonics:

Fitting the coefficients to observations, he showed that the field is overwhelmingly of internal origin. The Gauss coefficients \(g_n^m,h_n^m\) are still the standard description of the Earth’s field in the International Geomagnetic Reference Field. In honour of his absolute measurements, the CGS unit of magnetic flux density is called the gauss.

Survey, telegraph and politics, 1818–1840

In 1816 Gauss moved into the new observatory at Göttingen, where he lived for the rest of his life. His mother joined the household in 1817 and lived there until her death in 1839. In 1818 the Hanoverian government asked him to survey the kingdom and connect it to the Danish survey of Heinrich Christian Schumacher, a former student and lifelong friend. From 1821 to 1825 Gauss spent his summers in the field, measuring angles between church towers and hilltops. He invented the heliotrope, a mirror instrument that reflects sunlight to a distant station, to make long sightlines visible. The survey’s computations, by least squares, occupied him for years, and the geometric problems it raised fed directly into his 1828 memoir on surfaces.

His teaching load was light and he disliked lecturing to unprepared students, but he supervised several doctoral students who became important: the astronomer Christian Ludwig Gerling, the geometer Karl von Staudt, Johann Benedict Listing, a pioneer of topology, and later Richard Dedekind and Bernhard Riemann. August Ferdinand Möbius and Johann Franz Encke also studied with him.

Weber and the Göttingen Seven

In 1831 Gauss supported the appointment of the young physicist Wilhelm Weber. The two became close collaborators. In 1832 Gauss made the first absolute measurement of the Earth’s magnetic field. In 1833 they strung a wire about a kilometre long over the roofs of Göttingen between the observatory and the physics laboratory and sent messages along it, one of the first working electromagnetic telegraphs. In 1836 they founded the Göttingen Magnetic Union, a network of observatories making simultaneous measurements.

The collaboration was broken by politics. In 1837 the new King Ernst August of Hanover revoked the liberal constitution of 1833. Seven Göttingen professors protested and were dismissed, among them Weber and the orientalist Heinrich Ewald, husband of Gauss’s daughter Wilhelmine. Gauss, conservative by temperament, did not join the protest and could not reverse the dismissals. Weber returned to Göttingen in 1849. Gauss’s second wife Minna had died in 1831 after a long illness. In 1838 the Royal Society awarded him the Copley Medal.

Last years and legacy, 1840–1855

In his later years Gauss turned more to practical matters. He reorganized the Göttingen widows’ fund, studied actuarial questions and invested successfully in bonds. He learned Russian in his sixties in order to read Lobachevsky in the original. Farkas Bolyai had sent him the appendix of his son János Bolyai on non-Euclidean geometry, and in March 1832 Gauss replied that he could not praise it without praising himself, since it matched his own long-held views. The reply wounded the younger Bolyai.

In July 1849 Göttingen celebrated the fiftieth anniversary of his doctorate. For the occasion he gave his fourth proof of the fundamental theorem of algebra. Among the mathematicians present was Peter Gustav Lejeune Dirichlet. In 1851 he approved Riemann’s doctoral thesis, and in 1854 he chose, from three topics Riemann proposed for his habilitation lecture, the foundations of geometry. Riemann’s lecture of 10 June 1854 generalized Gauss’s intrinsic geometry of surfaces to any number of dimensions, and is said to have impressed the old man deeply.

Gauss’s health declined in 1854. His last outing was to see the opening of the railway from Hanover to Göttingen. He died in his sleep at the observatory on 23 February 1855, aged 77, and was buried in the Albani cemetery in Göttingen. Dirichlet was called from Berlin to succeed him.

Legacy

• Number theory: congruences, reciprocity laws, quadratic forms and class groups, cyclotomy and Gauss sums are the starting points of algebraic number theory, from Kummer and Dedekind to class field theory.
• Geometry: intrinsic curvature led through Riemann to modern differential geometry and to Einstein’s theory of gravity.
• Statistics and numerics: least squares, the normal distribution, Gaussian elimination, Gaussian quadrature and an early form of the fast Fourier transform, found in his work on asteroid orbits.
• Physics: Gauss’s law, absolute units, geomagnetic modelling; the CGS unit of magnetic flux density is named after him.
• Collected works: the Werke, in twelve volumes (1863–1933), include his diary and many unpublished papers.

Connections

People linked to Carl Friedrich Gauss across this collection. A name without a link has no page here yet. See Carl Friedrich Gauss in the network of all 337 lives →

Teachers and advisors

• Johann Friedrich Pfaff 1799

Students

• August Ferdinand Möbius
• Bernhard Riemann 1851
• Richard Dedekind 1852
• Christian Ludwig Gerling 1812
• Johann Benedict Listing 1834
• Johann Franz Encke
• Karl von Staudt 1822

Influenced by

• Daniel Bernoulli
• Jean le Rond d'Alembert
• Joseph-Louis Lagrange
• Leonhard Euler
• William Gilbert

Influenced

• Andrey Markov
• Carl Gustav Jacob Jacobi
• Eugenio Beltrami
• Gaston Darboux
• János Bolyai
• Niels Henrik Abel
• Peter Gustav Lejeune Dirichlet
• Robert Bunsen
• Sophie Germain 1804-1819
• Thomas Joannes Stieltjes
• Évariste Galois

Colleagues and collaborators

• Alexander von Humboldt 1828-1855
• Carl Gustav Jacob Jacobi
• Friedrich Wilhelm Bessel
• János Bolyai 1832
• Nikolai Lobachevsky 1840-1855
• Pierre-Simon Laplace
• Sophie Germain 1804-1808
• Wilhelm Weber 1831-1855

Rivals

• Adrien-Marie Legendre

Friends

• János Bolyai 1796-1855
• Heinrich Christian Schumacher
• Heinrich Wilhelm Olbers 1802-1840
• Martin Bartels

Family

• Heinrich Ewald

Also mentioned in this life

• Albert Einstein
• Euclid
• Pierre de Fermat
• Joseph Fourier
• Georg Frobenius
• Felix Klein
• Ernst Eduard Kummer
• James Clerk Maxwell
• Isaac Newton

Fields

• number theory 45 lives
• algebra 29 lives
• differential geometry 15 lives
• statistics 7 lives
• astronomy 50 lives
• geodesy 12 lives
• physics 13 lives

On CoursesHub

• Gauss · History of Mathematics
• Non-Euclidean Geometry · History of Mathematics & Physics
• Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Johann Carl Friedrich Gauss, MacTutor History of Mathematics.
2. Carl Friedrich Gauss, Wikipedia (with extensive references).
3. Gauss’s diary, Wikipedia.
4. W. K. Bühler, Gauss: A Biographical Study, Springer, 1981.
5. G. W. Dunnington, Carl Friedrich Gauss: Titan of Science, 1955; reissued by the MAA, 2004.
6. W. Sartorius von Waltershausen, Gauss zum Gedächtniss, Leipzig, 1856.

Mathematics

1. C. F. Gauss, Disquisitiones Arithmeticae, Leipzig, 1801; English translation by A. A. Clarke, Yale, 1966.
2. C. F. Gauss, Theoria motus corporum coelestium, Hamburg, 1809.
3. C. F. Gauss, Disquisitiones generales circa superficies curvas, 1828; English translation, General Investigations of Curved Surfaces, Dover.
4. C. F. Gauss, Werke, 12 vols., Göttingen, 1863–1933.
5. D. A. Cox, Primes of the Form \(x^2+ny^2\), Wiley, 2nd ed. 2013.
6. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1990.
7. J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
8. M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.
9. S. M. Stigler, The History of Statistics, Harvard, 1986.
10. F. Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein, Springer, 2000.