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Curl (mathematics)

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.

A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

The notation $curl F$ is more common in North America. In the rest of the world, particularly in 20th century scientific literature, the alternative notation $rot F$ is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator, as in $\nabla \times \mathbf{F}$, which also reveals the relation between curl (rotor), divergence, and gradient operators.

Unlike the gradient and divergence, curl as formulated in vector calculus does not generalize simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This deficiency is a direct consequence of the limitations of vector calculus; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensional cross product, and indeed the connection is reflected in the notation $$\nabla \times$$ for the curl.

The name "curl" was first suggested by James Clerk Maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839.

Definition
The curl of a vector field $F$, denoted by $r$, or $$\nabla \times \mathbf{F}$$, or $C$, is an operator that maps $C$ functions in $F$ to $curl F$ functions in $rot F$, and in particular, it maps continuously differentiable functions $C^{k}$ to continuous functions $R^{3}$. It can be defined in several ways, to be mentioned below:

One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if $$\mathbf{\hat{u}}$$ is any unit vector, the component of the curl of $C^{k−1}$ along the direction $$\mathbf{\hat{u}}$$ may be defined to be the limiting value of a closed line integral in a plane perpendicular to $$\mathbf{\hat{u}}$$ divided by the area enclosed, as the path of integration is contracted indefinitely around the point.

More specifically, the curl is defined at a point $R^{3}$ as $$(\nabla \times \mathbf{F})(p)\cdot \mathbf{\hat{u}} \ \overset{\underset{\mathrm{def}}{}}{{}={}} \lim_{A \to 0}\frac{1}{|A|}\oint_C \mathbf{F} \cdot \mathrm{d}\mathbf{r}$$ where the line integral is calculated along the boundary $R^{3} → R^{3}$ of the area $R^{3} → R^{3}$ in question, $F$ being the magnitude of the area. This equation defines the component of the curl of $p$ along the direction $$\mathbf{\hat{u}}$$. The infinitesimal surfaces bounded by $C$ have $$\mathbf{\hat{u}}$$ as their normal. $A$ is oriented via the right-hand rule.

The above formula means that the component of the curl of a vector field along a certain axis is the infinitesimal area density of the circulation of the field in a plane perpendicular to that axis. This formula does not a priori define a legitimate vector field, for the individual circulation densities with respect to various axes a priori need not relate to each other in the same way as the components of a vector do; that they do indeed relate to each other in this precise manner must be proven separately.

To this definition fits naturally the Kelvin–Stokes theorem, as a global formula corresponding to the definition. It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface.

Another way one can define the curl vector of a function $|A|$ at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing $F$ divided by the volume enclosed, as the shell is contracted indefinitely around $C$.

More specifically, the curl may be defined by the vector formula $$(\nabla \times \mathbf{F})(p) \overset{\underset{\mathrm{def}}{}}{{}={}} \lim_{V \to 0}\frac{1}{|V|}\oint_S \mathbf{\hat{n}} \times \mathbf{F} \ \mathrm{d}S$$ where the surface integral is calculated along the boundary $C$ of the volume $F$, $p$ being the magnitude of the volume, and $$\mathbf{\hat{n}}$$ pointing outward from the surface $p$ perpendicularly at every point in $S$.

In this formula, the cross product in the integrand measures the tangential component of $V$ at each point on the surface $|V|$, and points along the surface at right angles to the tangential projection of $S$. Integrating this cross product over the whole surface results in a vector whose magnitude measures the overall circulation of $S$ around $F$, and whose direction is at right angles to this circulation. The above formula says that the curl of a vector field at a point is the infinitesimal volume density of this "circulation vector" around the point.

To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the volume integral of the curl of a vector field to the above surface integral taken over the boundary of the volume.

Whereas the above two definitions of the curl are coordinate free, there is another "easy to memorize" definition of the curl in curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolic coordinates: $$\begin{align} & (\operatorname{curl}\mathbf F)_1=\frac{1}{h_2h_3}\left (\frac{\partial (h_3F_3)}{\partial u_2}-\frac{\partial (h_2F_2)}{\partial u_3}\right ), \\[5pt] & (\operatorname{curl}\mathbf F)_2=\frac{1}{h_3h_1}\left (\frac{\partial (h_1F_1)}{\partial u_3}-\frac{\partial (h_3F_3)}{\partial u_1}\right ), \\[5pt] & (\operatorname{curl}\mathbf F)_3=\frac{1}{h_1h_2}\left (\frac{\partial (h_2F_2)}{\partial u_1}-\frac{\partial (h_1F_1)}{\partial u_2}\right ). \end{align}$$

The equation for each component $S$ can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1 (where the subscripts represent the relevant indices).

If $F$ are the Cartesian coordinates and $F$ are the orthogonal coordinates, then $$h_i = \sqrt{\left (\frac{\partial x_1}{\partial u_i} \right )^2 + \left (\frac{\partial x_2}{\partial u_i} \right )^2 + \left (\frac{\partial x_3}{\partial u_i} \right )^2}$$ is the length of the coordinate vector corresponding to $S$. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1.

Usage
In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived.

The notation $(curl F)_{k}$ has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if $(x_{1}, x_{2}, x_{3})$ is taken as a vector differential operator del. Such notation involving operators is common in physics and algebra.

Expanded in 3-dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations),$(u_{1}, u_{2}, u_{3})$ is, for $u_{i}$ composed of $∇ × F$ (where the subscripts indicate the components of the vector, not partial derivatives): $$ \nabla \times \mathbf{F} = \begin{vmatrix} \boldsymbol{\hat\imath} & \boldsymbol{\hat\jmath} & \boldsymbol{\hat k} \\[5mu] {\dfrac{\partial}{\partial x}} & {\dfrac{\partial}{\partial y}} & {\dfrac{\partial}{\partial z}} \\[5mu] F_x & F_y & F_z \end{vmatrix} $$ where $∇$, $∇ × F$, and $F$ are the unit vectors for the $[F_{x}, F_{y}, F_{z}]$-, $i$-, and $j$-axes, respectively. This expands as follows: $$ \nabla \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \boldsymbol{\hat\imath} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \boldsymbol{\hat\jmath} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \boldsymbol{\hat k} $$

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.

In a general coordinate system, the curl is given by $$(\nabla \times \mathbf{F} )^k = \frac{1}{\sqrt{g}} \varepsilon^{k\ell m} \nabla_\ell F_m$$ where $ε$ denotes the Levi-Civita tensor, $k$ the covariant derivative, $$ g$$ is the determinant of the metric tensor and the Einstein summation convention implies that repeated indices are summed over. Due to the symmetry of the Christoffel symbols participating in the covariant derivative, this expression reduces to the partial derivative: $$(\nabla \times \mathbf{F} ) = \frac{1}{\sqrt{g}} \mathbf{R}_k\varepsilon^{k\ell m} \partial_\ell F_m$$ where $x$ are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as: $$ \nabla \times \mathbf{F} = \left( \star \big( {\mathrm d} \mathbf{F}^\flat \big) \right)^\sharp $$

Here ♭ and ♯ are the musical isomorphisms, and $y$ is the Hodge star operator. This formula shows how to calculate the curl of $z$ in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.

Example 1
Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The curl of the vector field at any point is given by the rotation of an infinitesimal area in the xy-plane (for z-axis component of the curl), zx-plane (for y-axis component of the curl) and yz-plane (for x-axis component of the curl vector). This can be seen in the examples below.

Example 2
The vector field $$\mathbf{F}(x,y,z)=y\boldsymbol{\hat{\imath}}-x\boldsymbol{\hat{\jmath}}$$ can be decomposed as $$F_x =y, F_y = -x, F_z =0.$$

Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linear force acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed.

Calculating the curl: $$\nabla \times \mathbf{F} =0\boldsymbol{\hat{\imath}}+0\boldsymbol{\hat{\jmath}}+ \left({\frac{\partial}{\partial x}}(-x) -{\frac{\partial}{\partial y}} y\right)\boldsymbol{\hat{k}}=-2\boldsymbol{\hat{k}} $$

The resulting vector field describing the curl would at all points be pointing in the negative $∇$ direction. The results of this equation align with what could have been predicted using the right-hand rule using a right-handed coordinate system. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed.

Example 3
For the vector field $$\mathbf{F}(x,y,z) = -x^2\boldsymbol{\hat{\jmath}}$$

the curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line $R_{k}$, the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the right-hand rule, it can be predicted that the resulting curl would be straight in the negative $★$ direction. Inversely, if placed on $F$, the object would rotate counterclockwise and the right-hand rule would result in a positive $F(x,y)=[y,−x]$ direction.

Calculating the curl: $${\nabla} \times \mathbf{F} = 0 \boldsymbol{\hat{\imath}} + 0\boldsymbol{\hat{\jmath}} + {\frac{\partial}{\partial x}}\left(-x^2\right) \boldsymbol{\hat{k}} = -2x\boldsymbol{\hat{k}}.$$

The curl points in the negative $z$ direction when $F(x, y) = [0, −x^{2}]$ is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane $x = 3$.

Further examples

 * In a vector field describing the linear velocities of each part of a rotating disk in uniform circular motion, the curl has the same value at all points, and this value turns out to be exactly two times the vectorial angular velocity of the disk (oriented as usual by the right-hand rule). More generally, for any flowing mass, the linear velocity vector field at each point of the mass flow has a curl (the vorticity of the flow at that point) equal to exactly two times the local vectorial angular velocity of the mass about the point.
 * For any solid object subject to an external physical force (such as gravity or the electromagnetic force), one may consider the vector field representing the infinitesimal force-per-unit-volume contributions acting at each of the points of the object. This force field may create a net torque on the object about its center of mass, and this torque turns out to be directly proportional and vectorially parallel to the (vector-valued) integral of the curl of the force field over the whole volume.
 * Of the four Maxwell's equations, two—Faraday's law and Ampère's law—can be compactly expressed using curl. Faraday's law states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and the time rate of change of the electric field.

Identities
In general curvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fields $z$ and $x = −3$ can be shown to be $$\nabla \times \left( \mathbf{v \times F} \right) = \Big( \left( \mathbf{ \nabla \cdot F } \right) + \mathbf{F \cdot \nabla} \Big) \mathbf{v}- \Big( \left( \mathbf{ \nabla \cdot v } \right) + \mathbf{v \cdot \nabla} \Big) \mathbf{F} \. $$

Interchanging the vector field $z$ and $z$ operator, we arrive at the cross product of a vector field with curl of a vector field: $$ \mathbf{v \ \times } \left( \mathbf{ \nabla \times F} \right) =\nabla_\mathbf{F} \left( \mathbf{v \cdot F } \right) - \left( \mathbf{v \cdot \nabla } \right) \mathbf{F} \, $$ where $x$ is the Feynman subscript notation, which considers only the variation due to the vector field $x = 0$ (i.e., in this case, $v$ is treated as being constant in space).

Another example is the curl of a curl of a vector field. It can be shown that in general coordinates $$ \nabla \times \left( \mathbf{\nabla \times F} \right) = \mathbf{\nabla}(\mathbf{\nabla \cdot F}) - \nabla^2 \mathbf{F} \, $$ and this identity defines the vector Laplacian of $F$, symbolized as $v$.

The curl of the gradient of any scalar field $φ$ is always the zero vector field $$\nabla \times ( \nabla \varphi ) = \boldsymbol{0}$$ which follows from the antisymmetry in the definition of the curl, and the symmetry of second derivatives.

The divergence of the curl of any vector field is equal to zero: $$\nabla\cdot(\nabla\times\mathbf{F}) = 0.$$

If $φ$ is a scalar valued function and $∇$ is a vector field, then $$\nabla \times ( \varphi \mathbf{F}) = \nabla \varphi \times \mathbf{F} + \varphi \nabla \times \mathbf{F} $$

Generalizations
The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra $$\mathfrak{so}(3)$$ of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and $\mathfrak{so}(3)$, these all being 3-dimensional spaces.

Differential forms
In 3 dimensions, a differential 0-form is a real-valued function $∇_{F}$; a differential 1-form is the following expression, where the coefficients are functions: $$a_1\,dx + a_2\,dy + a_3\,dz;$$ a differential 2-form is the formal sum, again with function coefficients: $$a_{12}\,dx\wedge dy + a_{13}\,dx\wedge dz + a_{23}\,dy\wedge dz;$$ and a differential 3-form is defined by a single term with one function as coefficient: $$a_{123}\,dx\wedge dy\wedge dz.$$ (Here the $a$-coefficients are real functions of three variables; the "wedge products", e.g. $F$, can be interpreted as some kind of oriented area elements, $v$, etc.)

The exterior derivative of a $F$-form in $∇^{2}F$ is defined as the $F$-form from above—and in $f(x, y, z)$ if, e.g., $$\omega^{(k)}=\sum_{1\leq i_1<i_2<\cdots<i_k\leq n} a_{i_1,\ldots,i_k} \,dx_{i_1}\wedge \cdots\wedge dx_{i_k},$$ then the exterior derivative $dx ∧ dy$ leads to $$ d\omega^{(k)}=\sum_{\scriptstyle{j=1} \atop \scriptstyle{i_1<\cdots<i_k}}^n\frac{\partial a_{i_1,\ldots,i_k}}{\partial x_j}\,dx_j \wedge dx_{i_1}\wedge \cdots \wedge dx_{i_k}.$$

The exterior derivative of a 1-form is therefore a 2-form, and that of a 2-form is a 3-form. On the other hand, because of the interchangeability of mixed derivatives, $$\frac{\partial^2}{\partial x_i\,\partial x_j} = \frac{\partial^2}{\partial x_j\,\partial x_i}, $$ and antisymmetry, $$d x_i \wedge d x_j = -d x_j \wedge d x_i$$

the twofold application of the exterior derivative yields $$0$$ (the zero $$k+2$$-form).

Thus, denoting the space of $dx ∧ dy = −dy ∧ dx$-forms by $k$ and the exterior derivative by $R^{3}$ one gets a sequence: $$0 \, \overset{d}{\longrightarrow} \; \Omega^0\left(\mathbb{R}^3\right) \, \overset{d}{\longrightarrow} \; \Omega^1\left(\mathbb{R}^3\right) \, \overset{d}{\longrightarrow} \; \Omega^2\left(\mathbb{R}^3\right) \, \overset{d}{\longrightarrow} \; \Omega^3\left(\mathbb{R}^3\right) \, \overset{d}{\longrightarrow} \, 0.$$

Here $(k + 1)$ is the space of sections of the exterior algebra $R^{n}$ vector bundle over Rn, whose dimension is the binomial coefficient $d$; note that $k$ for $Ω^{k}(R^{3})$ or $d$. Writing only dimensions, one obtains a row of Pascal's triangle:

the 1-dimensional fibers correspond to scalar fields, and the 3-dimensional fibers to vector fields, as described below. Modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div.

Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric. On a Riemannian manifold, or more generally pseudo-Riemannian manifold, $Ω^{k}(R^{n})$-forms can be identified with $Λ^{k}(R^{n})$-vector fields ($( n k )$-forms are $Ω^{k}(R^{3}) = 0$-covector fields, and a pseudo-Riemannian metric gives an isomorphism between vectors and covectors), and on an oriented vector space with a nondegenerate form (an isomorphism between vectors and covectors), there is an isomorphism between $k > 3$-vectors and $k < 0$-vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an oriented pseudo-Riemannian manifold, one can interchange $k$-forms, $k$-vector fields, $k$-forms, and $k$-vector fields; this is known as Hodge duality. Concretely, on $k$ this is given by:
 * 1-forms and 1-vector fields: the 1-form $(n − k)$ corresponds to the vector field $k$.
 * 1-forms and 2-forms: one replaces $k$ by the dual quantity $(n − k)$ (i.e., omit $(n − k)$), and likewise, taking care of orientation: $R^{3}$ corresponds to $a_{x} dx + a_{y} dy + a_{z} dz$, and $(a_{x}, a_{y}, a_{z})$ corresponds to $dx$. Thus the form $dy ∧ dz$ corresponds to the "dual form" $dx$.

Thus, identifying 0-forms and 3-forms with scalar fields, and 1-forms and 2-forms with vector fields:
 * grad takes a scalar field (0-form) to a vector field (1-form);
 * curl takes a vector field (1-form) to a pseudovector field (2-form);
 * div takes a pseudovector field (2-form) to a pseudoscalar field (3-form)

On the other hand, the fact that $dy$ corresponds to the identities $$\nabla\times(\nabla f) = \mathbf 0$$ for any scalar field $f$, and $$\nabla \cdot (\nabla \times\mathbf v)=0$$ for any vector field $dz ∧ dx = −dx ∧ dz$.

Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation, because the spaces of 0-forms and $dz$-forms at each point are always 1-dimensional and can be identified with scalar fields, while the spaces of 1-forms and $dx ∧ dy$-forms are always fiberwise $a_{x} dx + a_{y} dy + a_{z} dz$-dimensional and can be identified with vector fields.

Curl does not generalize in this way to 4 or more dimensions (or down to 2 or fewer dimensions); in 4 dimensions the dimensions are

so the curl of a 1-vector field (fiberwise 4-dimensional) is a 2-vector field, which at each point belongs to 6-dimensional vector space, and so one has $$\omega^{(2)}=\sum_{i<k=1,2,3,4}a_{i,k}\,dx_i\wedge dx_k,$$ which yields a sum of six independent terms, and cannot be identified with a 1-vector field. Nor can one meaningfully go from a 1-vector field to a 2-vector field to a 3-vector field (4 → 6 → 4), as taking the differential twice yields zero ($a_{z} dx ∧ dy + a_{y} dz ∧ dx + a_{x} dy ∧ dz$). Thus there is no curl function from vector fields to vector fields in other dimensions arising in this way.

However, one can define a curl of a vector field as a 2-vector field in general, as described below.

Curl geometrically
2-vectors correspond to the exterior power $d = 0$; in the presence of an inner product, in coordinates these are the skew-symmetric matrices, which are geometrically considered as the special orthogonal Lie algebra $v$ of infinitesimal rotations. This has $n$ dimensions, and allows one to interpret the differential of a 1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) we have $(n − 1)$, which is the most elegant and common case. In 2 dimensions the curl of a vector field is not a vector field but a function, as 2-dimensional rotations are given by an angle (a scalar – an orientation is required to choose whether one counts clockwise or counterclockwise rotations as positive); this is not the div, but is rather perpendicular to it. In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra $\mathfrak{so}(4)$.

The curl of a 3-dimensional vector field which only depends on 2 coordinates (say $n$ and $d = 0$) is simply a vertical vector field (in the $Λ^{2}V$ direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page.

Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus and associated physics to higher dimensions.

Inverse
In the case where the divergence of a vector field $$\mathfrak{so}$(V)$ is zero, a vector field $( n 2 )  = 1⁄2n(n − 1)$ exists such that $n = 1⁄2n(n − 1)$. This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential.

If $x$ is a vector field with $y$, then adding any gradient vector field $z$ to $V$ will result in another vector field $W$ such that $V = curl(W)$ as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknown irrotational field with the Biot–Savart law.

title: Curl (mathematics)


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Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß ; Carolus Fridericus Gauss; 30 April 1777 – 23 February 1855) was a German mathematician, astronomer, geodesist, and physicist who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855.

While studying at the University of Göttingen, he propounded several mathematical theorems. Gauss completed his masterpieces Disquisitiones Arithmeticae and Theoria motus corporum coelestium as a private scholar. He gave the second and third complete proofs of the fundamental theorem of algebra, made contributions to number theory, and developed the theories of binary and ternary quadratic forms.

Gauss was instrumental in the identification of Ceres as a dwarf planet. His work on the motion of planetoids disturbed by large planets led to the introduction of the Gaussian gravitational constant and the method of least squares, which he had discovered before Adrien-Marie Legendre published it.

Gauss was in charge of the extensive geodetic survey of the Kingdom of Hanover together with an arc measurement project from 1820 to 1844; he was one of the founders of geophysics and formulated the fundamental principles of magnetism. Fruits of his practical work were the inventions of the heliotrope in 1821, a magnetometer in 1833 and – alongside Wilhelm Eduard Weber – the first electromagnetic telegraph in 1833.

Gauss refused to publish incomplete work and left several works to be edited posthumously. He believed that the act of learning, not possession of knowledge, provided the greatest enjoyment. Gauss confessed to disliking teaching, but some of his students became influential mathematicians.

Youth and education
Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig) in the Duchy of Brunswick-Wolfenbüttel (now part of Germany's federal state Lower Saxony), to a family of lower social status. His father Gebhard Dietrich Gauss (1744–1808) worked in several jobs, as butcher, bricklayer, gardener, and as treasurer of a death-benefit fund. Gauss characterized his father as honourable and respected, but rough and dominating at home. He was experienced in writing and calculating, whereas his second wife Dorothea (1743–1839), Carl Friedrich's mother, was nearly illiterate. He had one elder brother from his father's first marriage.

Gauss was a child prodigy in mathematics. When the elementary teachers noticed his intellectual abilities, they brought him to the attention of the Duke of Brunswick who sent him to the local Collegium Carolinum, which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers. Thereafter the Duke granted him the resources for studies of mathematics, sciences, and classical languages at the Hanoverian University of Göttingen until 1798. His professor in mathematics was Abraham Gotthelf Kästner, whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians" because of his epigrams. Astronomy was taught by Karl Felix Seyffer, with whom Gauss stayed in correspondence after graduation; Olbers and Gauss mocked him in their correspondence. On the other hand, he thought highly of Georg Christoph Lichtenberg, his teacher of physics, and of Christian Gottlob Heyne, whose lectures in classics Gauss attended with pleasure. Fellow students of this time were Johann Friedrich Benzenberg, Farkas Bolyai, and Heinrich Wilhelm Brandes.

He was likely a self-taught student in mathematics since he independently rediscovered several theorems. He solved a geometrical problem that had occupied mathematicians since the days of the Ancient Greeks, when he determined in 1796 which regular polygons can be constructed by compass and straightedge. This discovery ultimately lead Gauss to choose mathematics instead of philology as a career. Gauss's mathematical diary, a collection of short remarks about his results from the years 1796 until 1814, shows that many ideas for his mathematical magnum opus Disquisitiones Arithmeticae (1801) date from this time.

Private scholar
Gauss graduated as a Doctor of Philosophy in 1799, not in Göttingen, as is sometimes stated, but at the Duke of Brunswick's special request from the University of Helmstedt, the only state university of the duchy. Johann Friedrich Pfaff assessed his doctoral thesis, and Gauss got the degree in absentia without further oral examination. The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss subsequently refused calls from the Russian Academy of Sciences in St. Peterburg and Landshut University. Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. Architect Peter Joseph Krahe made preliminary designs, but one of Napoleon's wars cancelled those plans: the Duke was killed in the battle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped.

When Gauss was calculating asteroid orbits in the first years of the century, he established contact with the astronomical community of Bremen and Lilienthal, especially Wilhelm Olbers, Karl Ludwig Harding, and Friedrich Wilhelm Bessel, as part of the informal group of astronomers known as the Celestial police. One of their aims was the discovery of further planets, and they assembled data on asteroids and comets as a basis for Gauss's research on their orbits, which he later published in his astronomical magnum opus Theoria motus corporum coelestium (1809).

Professor in Göttingen
In November 1807, Gauss followed a call to the University of Göttingen, then an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte, as full professor and director of the astronomical observatory, and kept the chair until his death in 1855. He was soon confronted with the demand for two thousand francs from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and Laplace wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person from Frankfurt, later discovered to be Prince-primate Dalberg, paid the sum.

Gauss took on the directorate of the 60-year-old observatory, founded in 1748 by Prince-elector George II and built on a converted fortification tower, with usable, but partly out-of-date instruments. The construction of a new observatory had been approved by Prince-elector George III in principle since 1802, and the Westphalian government continued the planning, but Gauss could not move to his new place of work until September 1816. He got new up-to-date instruments, including two meridian circles from Repsold and Reichenbach, and a heliometer from Fraunhofer.

The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: astronomy was the main focus in the first two decades of the 19th century, geodesy in the third decade, and physics, mainly magnetism, in the fourth decade.

Gauss made no secret of his aversion to giving academic lectures. But from the start of his academic career at Göttingen, he continuously gave lectures until 1854. He often complained about the burdens of teaching, feeling that it was a waste of his time. On the other hand, he occasionally described some students as talented. Most of his lectures dealt with astronomy, geodesy, and applied mathematics, and only three lectures on subjects of pure mathematics. Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: Moritz Cantor, Dedekind, Dirksen, Encke, Gould, Heine, Klinkerfues, Kupffer, Listing, Möbius, Nicolai, Riemann, Ritter, Schering, Scherk, Schumacher, von Staudt, Stern, Ursin; as geoscientists Sartorius von Waltershausen, and Wappäus.

Gauss did not write any textbook and disliked the popularization of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essay Erdmagnetismus und Magnetometer of 1836. Gauss published his papers and books exclusively in Latin or German. He wrote Latin in a merely classical style but used some customary modifications set by contemporary mathematicians.

In his inaugural lecture at Göttingen University from 1808, Gauss claimed reliable observations and results attained only by a strong calculus as the sole tasks of astronomy. At university, he was accompanied by a staff of other lecturers in his disciplines, who completed the educational program: for instance the mathematician Thibaut with his lectures, the physicist Mayer, known for his textbooks, his successor Weber since 1831, and in the observatory Harding, who took the main part of lectures in practical astronomy. When the observatory was completed, Gauss took his living accommodation in the western wing of the new observatory and Harding in the eastern one. They had once been on friendly terms with one another, but over time they became alienated, possibly – as some biographers presume – because Gauss had wished the equal-ranked Harding to be no more than his assistant or observer. Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except for some very seldom joint observations.

Brendel subdivides Gauss's astronomic activity chronologically into seven periods, of which the years since 1820 are taken as a "period of lower astronomical activity". The new, well-equipped observatory did not work as effectively as other ones; Gauss's astronomical research had the character of a one-man enterprise without a long-time observation program, and the university established a place for an assistant only after Harding died in 1834.

Nevertheless, Gauss twice refused the opportunity to solve the problem by accepting offers from Berlin in 1810 and 1825 to become a full member of the Prussian Academy without burdening lecturing duties, as well as from Leipzig University in 1810 and from Vienna University in 1842, perhaps because of the family's difficult situation. Gauss's salary was raised from 1000 Reichsthaler in 1810 to 2400 Reichsthaler in 1824, and in his later years, he was one of the best-paid professors of the university.

When Gauss was asked for help by his colleague and friend Friedrich Wilhelm Bessel in 1810, who was in trouble at Königsberg University because of his lack of an academic title, Gauss provided a doctorate honoris causa for Bessel from the Philosophy Faculty of Göttingen in March 1811. Gauss gave another recommendation for an honorary degree for Sophie Germain but only shortly before her death, so she never received it. He also gave successful support to the mathematician Gotthold Eisenstein in Berlin.

Gauss was loyal to the House of Hanover. After King William IV died in 1837, the personal union between the kingdoms of Great Britain and Ireland and Hanover ceased. In the same year, the new Hanoverian King Ernest Augustus annulled the 1833 constitution. Seven professors, later known as the "Göttingen Seven", protested against this, among them his friend and collaborator Wilhelm Weber and Gauss's son-in-law Heinrich Ewald. All of them were dismissed, and three of them were expelled, but Ewald and Weber could stay in Göttingen. Gauss was deeply affected by this quarrel but saw no possibility to help them.

Gauss took part in academic administration: three times he was elected as dean of the Faculty of Philosophy. Being entrusted with the widow's pension fund of the university, he dealt with actuarial science and wrote a report on the strategy for stabilizing the benefits. He was appointed director of the Royal Academy of Sciences in Göttingen for nine years, even in his last year of life.

Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. On 23 February 1855, he died of a heart attack in Göttingen; and was interred in the Albani Cemetery there. Heinrich Ewald, Gauss's son-in-law, and Wolfgang Sartorius von Waltershausen, Gauss's close friend and biographer, gave eulogies at his funeral.

Gauss was a successful investor and accumulated considerable wealth with stocks and securities, finally a value of more than 150 thousand Thaler; after his death, about 18 thousand Thaler were found hidden in his rooms.

Gauss's brain
The day after Gauss's death his brain was removed, preserved, and studied by Rudolf Wagner, who found its mass to be slightly above average, at 1492 g. The cerebral area was determined by Wagner's son Hermann in his doctoral thesis to be 219588 mm2. In 2013, a neurobiologist at the Max Planck Institute for Biophysical Chemistry in Göttingen discovered that Gauss's brain had been mixed up soon after the first investigations, due to mislabelling, with that of the physician Conrad Heinrich Fuchs, who died in Göttingen a few months after Gauss. A further investigation showed no remarkable anomalies in the brains of both persons. Thus, all investigations on Gauss's brain until 1998, except the first ones of Rudolf and Hermann Wagner, actually refer to the brain of Fuchs.

Family
Gauss married Johanna Osthoff (1780–1809) on 9 October 1805 in St. Catherine's church in Brunswick. They had two sons and one daughter: Joseph (1806–1873), Wilhelmina (1808–1840), and Louis (1809–1810). Johanna died on 11 October 1809, one month after the birth of Louis, who himself died a few months later. Gauss chose the first names of his children in honour of Giuseppe Piazzi, Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids.

On 4 August 1810, the widower married Wilhelmine (Minna) Waldeck (1788–1831), a friend of his first wife, with whom he had three more children: Eugen (later: Eugene) (1811–1896), Wilhelm (later: William) (1813–1879), and Therese (1816–1864). Minna Gauss died on 12 September 1831 after being seriously ill for more than a decade. Therese then took over the household and cared for Gauss for the rest of his life; after her father's death, she married actor Constantin Staufenau. Her sister Wilhelmina married the orientalist Heinrich Ewald. Gauss's mother Dorothea lived in his house from 1817 until she died in 1839.

The eldest son Joseph, whilst still a schoolboy, helped his father as an assistant during the survey campaign in the summer of 1821. After a short time at university, in 1824 Joseph joined the Hanoverian army and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network to the western parts of the kingdom. With his geodetical qualifications, he left the service and engaged in the construction of the railway network as director of the Royal Hanoverian State Railways. In 1836 he studied the railroad system in the US for some months.

Eugen left Göttingen in September 1830 and emigrated to the United States, where he joined the army for five years. He then worked for the American Fur Company in the Midwest. Later, he moved to Missouri and became a successful businessman. Wilhelm married a niece of the astronomer Bessel; then moved to Missouri, too, started as a farmer and became wealthy in the shoe business in St. Louis in later years. Eugene and William have numerous descendants in America, but the Gauss descendants left in Germany all derive from Joseph, as the daughters had no children.

The scholar
In the first two decades of the 19th century, Gauss was the only important mathematician in Germany, comparable to the leading French ones; his Disquisitiones Arithmeticae was the first mathematical book from Germany to be translated into the French language.

Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration. Whereas previous mathematicians like Leonhard Euler let the readers take part in their reasoning for new ideas, including certain erroneous deviations from the correct path, Gauss however introduced a new style of direct and complete explanation that did not attempt to show the reader the author's train of thought.

"Gauss was the first to restore that rigor of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of the preceding period in new developments."

But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows: "It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again."

The posthumous papers, his scientific diary, and short glosses in his own textbooks show that he worked to a great extent in an empirical way. He was a lifelong busy and enthusiastic calculator, who made his calculations with extraordinary rapidity, mostly without precise controlling, but checked the results by masterly estimation. Nevertheless, his calculations were not always free from mistakes. He coped with the enormous workload by using skillful tools. Gauss used a lot of mathematical tables, examined their exactness, and constructed new tables on various matters for personal use. He developed new tools for effective calculation, for example the Gaussian elimination. It has been taken as a curious feature of his working style that he carried out calculations with a high degree of precision much more than required, and prepared tables with more decimal places than ever requested for practical purposes. Very likely, this method gave him a lot of material which he used in finding theorems in number theory.

Gauss refused to publish work that he did not consider complete and above criticism. This perfectionism was in keeping with the motto of his personal seal Pauca sed Matura ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself, claiming that the initial discovery of ideas was easy, but preparing a presentable elaboration was a demanding matter for him, for either lack of time or "serenity of mind". Nevertheless, he published many short communications of urgent content in various journals, but left a considerable literary estate, too. Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.

On certain occasions, Gauss claimed that the ideas of another scholar had already been in his possession previously. Thus his concept of priority as "the first to discover, not the first to publish" differed from that of his scientific contemporaries. In contrast to his perfectionism in presenting mathematical ideas, he was criticized for a negligent way of quoting. He justified himself with a very special view of correct quoting: if he gave references, then only in a quite complete way, with respect to the previous authors of importance, which no one should ignore; but quoting in this way needed knowledge of the history of science and more time than he wished to spend.

The private man
Soon after Gauss's death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw him as a serene and forward-striving man with childlike modesty, but also of "iron character" with an unshakeable strength of mind. Apart from his closer circle, others regarded him as reserved and unapproachable "like an Olympian sitting enthroned on the summit of science". His close contemporaries agreed that Gauss was a man of difficult character. He often refused to accept compliments. His visitors were occasionally irritated by his grumpy behaviour, but a short time later his mood could change, and he would become a charming, open-minded host. Gauss abominated polemic natures; together with his colleague Hausmann he opposed to a call for Justus Liebig on a university chair in Göttingen, "because he was always involved in some polemic."

Gauss's life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the birth of their third child, he revealed the grief in a last letter to his dead wife in the style of an ancient threnody, the most personal surviving document of Gauss. The situation worsened when tuberculosis afflicted and ultimately destroyed the health of his second wife Minna over 13 years; both his daughters later suffered from the same disease. Gauss himself gave only slight hints of his distress: in a letter to Bessel dated December 1831 he described himself as "the victim of the worst domestic sufferings".

By reason of his wife's illness, both younger sons were educated for some years in Celle far from Göttingen. The military career of his elder son Joseph ended after more than two decades with the rank of a poorly paid first lieutenant, although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married. The second son Eugen shared a good measure of his father's talent in computation and languages, but had a vivacious and sometimes rebellious character. He wanted to study philology, whereas Gauss wanted him to become a lawyer. Having run up debts and caused a scandal in public, Eugen suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. He wasted the little money he had taken to start, after which his father refused further financial support. The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties getting an appropriate education, and eventually emigrated as well. Only Gauss's youngest daughter Therese accompanied him in his last years of life.

Collecting numerical data on very different things, useful or useless, became a habit in his later years, for example, the number of paths from his home to certain places in Göttingen, or the number of living days of persons; he congratulated Humboldt in December 1851 for having reached the same age as Isaac Newton at his death, calculated in days.

Similar to his excellent knowledge of Latin he was also acquainted with modern languages. At the age of 62, he began to teach himself Russian, very likely to understand scientific writings from Russia, among them those of Lobachevsky on non-Euclidean geometry. Gauss read both classical and modern literature, and English and French works in the original languages. His favorite English author was Walter Scott, his favorite German Jean Paul. Gauss liked singing and went to concerts. He was a busy newspaper reader; in his last years, he used to visit an academic press salon of the university every noon. Gauss did not care much for philosophy, and mocked the "splitting hairs of the so-called metaphysicians", by which he meant proponents of the contemporary school of Naturphilosophie.

Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, following the motto "mundus vult decipi". He disliked Napoleon and his system, and all kinds of violence and revolution caused horror to him. Thus he condemned the methods of the Revolutions of 1848, though he agreed with some of their aims, such as the idea of a unified Germany. As far as the political system is concerned, he had a low estimation of the constitutional system; he criticized parliamentarians of his time for a lack of knowledge and logical errors.

Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes" and "I succeeded - not on account of my hard efforts, but by the grace of the Lord." Gauss was a member of the Lutheran church, like most of the population in northern Germany. It seems that he did not believe all dogmas or understand the Holy Bible quite literally. Sartorius mentioned Gauss's religious tolerance, and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions.

Fundamental theorem of algebra
In his doctoral thesis from 1799, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.

Disquisitiones Arithmeticae
In the preface to the Disquisitiones, Gauss dates the beginning of his work on number theory to the year 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had discovered by himself. The Disquisitiones Arithmeticae, written since 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraic number theory. Therein he introduces the triple bar symbol ($≡$) for congruence and uses it for a clean presentation of modular arithmetic. It deals with the unique factorization theorem and primitive roots modulo n. In the main chapters, Gauss presents the first two proofs of the law of quadratic reciprocity and develops the theories of binary and ternary quadratic forms.

The Disquisitiones include the Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his theorem on three squares, he proves the triangular case of the Fermat polygonal number theorem for n = 3. From several analytic results on class numbers that Gauss gives without proof towards the end of the fifth chapter, it appears that Gauss already knew the class number formula in 1801.

In the last chapter, Gauss gives proof for the constructibility of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical problem to an algebraic one. He shows that a regular polygon is constructible if the number of its sides is either a power of 2 or the product of a power of 2 and any number of distinct Fermat primes. In the same chapter, he gives a result on the number of solutions of certain cubic polynomials with coefficients in finite fields, which amounts to counting integral points on an elliptic curve. An unfinished eight chapter was found among left papers only after his death, consisting of work done during the years 1797–1799.

Further investigations
One of Gauss's first results was the empirically found conjecture of 1792 – the later called prime number theorem – giving an estimation of the number of prime numbers by using the integral logarithm.

When Olbers encouraged Gauss in 1816 to compete for a prize from the French Academy on proof for Fermat's Last Theorem (FLT), he refused because of his low esteem on this matter. However, among his left works a short undated paper was found with proofs of FLT for the cases n = 3 and n = 5. The particular case of n = 3 was proved much earlier by Leonhard Euler, but Gauss developed a more streamlined proof which made use of Eisenstein integers; though more general, the proof was simpler than in the real integers case.

Gauss contributed to solving the Kepler conjecture in 1831 with the proof that a greatest packing density of spheres in the three-dimensional space is given when the centers of the spheres form a cubic face-centered arrangement, when he reviewed a book of Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms. Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements.

In two papers on biquadratic residues (1828, 1832) Gauss introduced the ring of Gaussian integers $$\mathbb{Z}[i]$$, showed that it is a unique factorization domain. and generalized some key arithmetic concepts, such as Fermat's little theorem and Gauss's lemma. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.

In the second paper, he stated the general law of biquadratic reciprocity and proved several special cases of it. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claimed the techniques of these proofs (Gauss sums) can be applied to prove higher reciprocity laws.

Analysis
One of Gauss's first discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers. He discovered its relation to elliptic integrals in the years 1798–1799 through the Landen's transformation, and a diary entry recorded the discovery of the connection of Gauss's constant to lemniscatic elliptic functions, a result that Gauss stated that "will surely open an entirely new field of analysis". He also made early inroads into the more formal issues of the foundations of complex analysis, and from a letter to Bessel in 1811 it is clear that he knew the "fundamental theorem of complex analysis" – Cauchy's integral theorem – and understood the notion of complex residues when integrating around poles.

Euler's pentagonal numbers theorem, together with other researches on the AGM and lemniscatic functions, led him to plenty of results on Jacobi theta functions, culminating in the discovery in 1808 of the later called Jacobi triple product identity, which includes Euler's theorem as a special case. His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808.

Several mathematical fragments in his Nachlass indicate that he knew parts of the modern theory of modular forms. In his work on the multivalued AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM to its two "simplest values". In his unpublished writings he recognized and made a sketch of the key concept of fundamental domain for the modular group. One of Gauss's sketches of this kind was a drawing of a tessellation of the unit disk by "equilateral" hyperbolic triangles with all angles equal to $$\pi/4$$.

An example of Gauss's insight in the fields of analysis is the cryptic remark that the principles of circle division by compass and straightedge can also be applied to the division of the lemniscate curve, which inspired Abel's theorem on lemniscate division. Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of quadratic Gauss sum, in which he solved the main problem by introducing q-analogs of binomial coefficients and manipulating them by several original identities that seem to stem out of his work on elliptic functions theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic functions theory, and only the later work of mathematicians such as Jacobi and Hermite has exposed the crux of his argument.

In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general hypergeometric function $$F(\alpha,\beta,\gamma,x)$$, and shows that many of the functions known at the time are special cases of the hypergeometric function. This work is the first one with an exact inquiry of convergence of infinite series in the history of mathematics. Furthermore, it deals with infinite continued fractions arising as ratios of hypergeometric functions which are now called Gauss continued fractions.

In 1823, Gauss won the prize of the Danish Society with an essay on conformal mappings, which contains several developments that pertain to the field of complex analysis. Gauss stated that angle-preserving mappings in the complex plane must be complex analytic functions, and used the later called Beltrami equation to prove the existence of isothermal coordinates on analytic surfaces. The essay concludes with examples of conformal mappings into a sphere and an ellipsoid of revolution.

Numeric analysis
Gauss often deduced theorems inductively from numerical data he had collected empirically. As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions to numeric analysis, as the method of Gaussian quadrature published in 1816.

In a private letter to Gerling from 1823, he described a solution of a certain 4X4 system of linear equations by using Gauss-Seidel method – an "indirect" iterative method for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than 2 equations.

Gauss invented an algorithm for calculating what is now called discrete Fourier transforms, when calculating the orbits of Pallas and Juno in 1805, 160 years before Cooley and Tukey found their similar Cooley–Tukey FFT algorithm. He developed it as a trigonometric interpolation method, but the paper Theoria Interpolationis Methodo Nova Tractata was published only posthumously in 1876, preceded by the first presentation by Joseph Fourier on the subject in 1807.

Chronology
The first publication following the doctoral thesis dealt with the determination of the date of Easter (1800), a very elementary matter of mathematics. Gauss aimed to present a most convenient algorithm for people without any knowledge of ecclesiastical or even astronomical chronology, and thus avoided the usually required terms of golden number, epact, solar cycle, domenical letter, and any religious connotations. Biographers speculated on the reason why Gauss dealt with this matter, but it is likely comprehensible by the historical background. The replacement of the Julian calendar by the Gregorian calendar had caused confusion to the hundreds of states of the Holy Roman Empire since the 16th century, and was finished in Germany not until the year 1700, when the difference of eleven days was deleted, but the difference in calculating the date of Easter remained between Protestant and Catholic territories. A further agreement of 1776 equalized the confessional way of counting, thus in the Protestant states like the Duchy of Brunswick the Easter of 1777, five weeks before Gauss's birth, was the first one calculated in the new manner. The public difficulties of replacement may be the historical background for the confusion on this matter in the Gauss family (see chapter: Anecdotes). For being connected with the Easter regulations, an essay on the date of Pesach followed soon in 1802.

Astronomy
On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered a new celestial object, presumed it to be the long searched planet between Mars and Jupiter according to the so-called Titius–Bode law, and named it Ceres. He could track it only for a short time until it disappeared behind the glare of the Sun. The mathematical tools of the time were not sufficient to extrapolate a position from the few data for its reappearance. Gauss tackled the problem and predicted a position for possible rediscovery in December 1801. This turned out to be accurate within a half-degree when Franz Xaver von Zach on 7 and 31 December at Gotha, and independently Heinrich Olbers on 1 and 2 January in Bremen, identified the object near the predicted position.

Gauss's method leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work, Gauss used comprehensive approximation methods which he created for that purpose.

The discovery of Ceres led Gauss to the theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum. It introduced the Gaussian gravitational constant.

Since the new asteroids had been discovered, Gauss occupied himself with the perturbations of their orbital elements. Firstly he examined Ceres with analytical methods similar to those of Laplace, but his favorite object was Pallas, because of its great eccentricity and orbital inclination, whereby Laplace's method did not work. Gauss used his own tools: the arithmetic–geometric mean, the hypergeometric function, and his method of interpolation. He found an orbital resonance with Jupiter in proportion 18:7 in 1812; Gauss gave this result as cipher, and gave the explicit meaning only in letters to Olbers and Bessel. After long years of work, he finished it in 1816 without a result that seemed sufficient to him. This marked the end of his activities in theoretical astronomy, too.

One fruit of Gauss's research on Pallas perturbations was the Determinatio Attractionis... (1818) on a method of theoretical astronomy that later became known as the "elliptic ring method". It introduced an averaging conception in which a planet in orbit is replaced by a fictitious ring with mass density proportional to the time taking the planet to follow the corresponding orbital arcs. Gauss presents the method of evaluating the gravitational attraction of such an elliptic ring, which includes several steps; one of them involves a direct application of the arithmetic-geometric mean (AGM) algorithm to calculate an elliptic integral.

While Gauss's contributions to theoretical astronomy came to a marked end, more practical activities in observational astronomy continued and occupied him during his entire career. Even early in 1799, Gauss dealt with the determination of longitude by use of the lunar parallax, for which he developed more convenient formulas than those were in common use. After appointment as director of observatory he attached importance to the fundamental astronomical constants in correspondence with Bessel. Gauss himself provided tables for nutation and aberration, the solar coordinates, and refraction. He made many contributions to spherical geometry, and in this context solved some practical problems about navigation by stars. He published a great number of observations, mainly on minor planets and comets; his last observation was the solar eclipse of July 28, 1851.

Theory of errors
Gauss likely used the method of least squares for calculating the orbit of Ceres to minimize the impact of measurement error. The method was published first by Adrien-Marie Legendre in 1805, but Gauss claimed in Theoria motus (1809) that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares". Gauss proved that the method has the lowest sampling variance within the class of linear unbiased estimators under the assumption of normally distributed errors (Gauss–Markov theorem), in the two-part paper Theoria combinationis observationum erroribus minimis obnoxiae (1823).

In the first paper he proved Gauss's inequality (a Chebyshev-type inequality) for unimodal distributions, and stated without proof another inequality for moments of the fourth order (a special case of Gauss-Winckler inequality). He derived lower and upper bounds for the variance of sample variance. In the second paper, Gauss described recursive least squares methods. His work on the theory of errors was extended in several directions by the geodesist Friedrich Robert Helmert to the Gauss-Helmert model.

Gauss also contributed to problems in probability theory that are not directly concerned with the theory of errors. One example appears as a diary note where he tried to describe the asymptotic distribution of entries in the continued fraction expansion of a random number uniformly distributed in (0,1). He derived this distribution, now known as the Gauss-Kuzmin distribution, as a by-product of the discovery of the ergodicity of the Gauss map for continued fractions. Gauss's solution is the first-ever result in the metrical theory of continued fractions.

Arc measurement and geodetic survey
Gauss was busy with geodetic problems since 1799 when he helped Karl Ludwig von Lecoq with calculations during his survey in Westphalia. Later since 1804, he taught himself some geodetic practise with a sextant in Brunswick, and Göttingen.

Since 1816, Gauss's former student Heinrich Christian Schumacher, then professor in Copenhagen, but living in Altona (Holstein) near Hamburg as head of an observatory, carried out a triangulation of the Jutland peninsula from Skagen in the north to Lauenburg in the south. This project was the basis for map production but also aimed at determining the geodetic arc between the terminal sites. Data from geodetic arcs were used to determine the dimensions of the earth geoid, and long arc distances brought more precise results. Schumacher asked Gauss to continue this work further to the south in the Kingdom of Hanover; Gauss agreed after a short time of hesitation. Finally, in May 1820, King George IV gave the order to Gauss.

An arc measurement needs a precise astronomical determination of at least two points in the network. Gauss and Schumacher used the favourite occasion that both observatories in Göttingen and Altona, in the garden of Schumacher's house, laid nearly in the same longitude. The latitude was measured with both their instruments and a zenith sector of Ramsden that was transported to both observatories.

Gauss and Schumacher had already determined some angles between Lüneburg, Hamburg, and Lauenburg for the geodetic connection in October 1818. During the summers of 1821 until 1825 Gauss directed the triangulation work personally, that reached from Thuringia in the south to the river Elbe in the north. The triangle between Hoher Hagen, Großer Inselsberg in the Thuringian Forest, and Brocken in the Harz mountains was the largest one Gauss had ever measured with a maximum size of 107 km. In the thinly populated Lüneburg Heath without significant natural summits or artificial buildings, he had difficulties to find suitable triangulation points, sometimes cutting lanes through the vegetation was necessary.

For pointing signals, Gauss invented a new instrument with movable mirrors and a small telescope that reflects the sunbeams to the triangulation points, and named it heliotrope. Another suitable construction for the same purpose was a sextant with an additional mirror which he named vice heliotrope. Gauss got assistance by soldiers of the Hanoverian army, among them his eldest son Joseph. Gauss took part in the baseline measurement (Braak Base Line) of Schumacher in the village of Braak near Hamburg in 1820, and used the result for the evaluation of the Hanoverian triangulation.

An additional result was a better value of flattening of the approximative Earth ellipsoid. Gauss developed the universal transverse Mercator projection of the ellipsoidal shaped Earth (what he named conform projection) for representing geodetical data in plane charts.

When the arc measurement was finished, Gauss began the enlargement of the triangulation to the west to get a survey of the whole Kingdom of Hanover with a Royal decree from 25 March 1828. The practical work was directed by three army officers, among them Lieutenant Joseph Gauss. The complete data evaluation laid in the hands of Gauss, who applied his mathematical inventions such as the method of least squares and the elimination method to it. The project was finished in 1844, and Gauss sent a final report of the project to the government; his method of projection was not edited until 1866.

In 1828, when studying differences in latitude, Gauss first defined a physical approximation for the figure of the Earth as the surface everywhere perpendicular to the direction of gravity; later his doctoral student Johann Benedict Listing called this the geoid.

Differential geometry
The geodetic survey of Hanover fueled Gauss's interest in differential geometry and topology, fields of mathematics dealing with curves and surfaces. This led him in 1828 to the publication of a memoir that marks the birth of modern differential geometry of surfaces, as it departed from the traditional ways of treating surfaces as cartesian graphs of functions of two variables, and that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. As a result, the Theorema Egregium (remarkable theorem), established a property of the notion of Gaussian curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, regardless of the embedding of the surface in 3-dimensional or 2-dimensional space.

The Theorema Egregium leads to the abstraction of surfaces as doubly-extended manifolds; it clarifies the distinction between the intrinsic properties of the manifold (the metric) and its physical realization (the embedding) in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a sphere or an ellipsoid cannot be transformed to a plane without distortion, which causes a fundamental problem in designing projections for geographical maps. A portion of this essay is dedicated to a profound study of geodesics. In particular, Gauss proves the local Gauss-Bonnet theorem on geodesic triangles, and generalizes Legendre's theorem on spherical triangles to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the surface curvature at the vertices of the triangle, regardless of the behaviour of the surface in the triangle interior.

Gauss's memoir from 1828 lacks the conception of geodesic curvature. However, in a previously unpublished manuscript, very likely written in the years 1822 until 1825, he introduced the term "side curvature" (in German: "Seitenkrümmung") and proved its invariance under isometric transformations, a result that was later obtained by Ferdinand Minding and published by him in 1830. This Gauss paper contains the core of his lemma on total curvature, but also its generalization, found and proved by Pierre Ossian Bonnet in 1848 and known as Gauss-Bonnet theorem.

Non-Euclidean geometry
In the lifetime of Gauss, a vivid discussion on the Parallel postulate in Euclidean geometry was going on. Numerous efforts were made to prove it in the frame of the Euclidean axioms, whereas some mathematicians discussed the possibility of geometrical systems without it. Gauss thought about the basics of geometry since the 1790s years, but in the 1810s years, he realized that a non-Euclidean geometry without the parallel postulate could solve the problem. In a letter to Franz Taurinus of 1824, he presented a short comprehensible outline of what he named a "non-Euclidean geometry", but he strongly forbade Taurinus to make any use of it.

The first publications on non-Euclidean geometry in the history of mathematics were authored by Nikolai Lobachevsky in 1829 and Janos Bolyai in 1832. In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion. Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai claiming that these were congruent to his own thoughts since some decades. However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, for he his letter remarks are only vague and obscure.

Sartorius mentioned Gauss's work on non-Euclidean geometry firstly in 1856, but only the edition of left papers in Volume VIII of the Collected Works (1900) showed Gauss's ideas on that matter, at a time when non-Euclidean geometry had yet grown out of controversial discussion.

Early topology
Gauss was also an early pioneer of topology or Geometria Situs, as it was called in his lifetime. The first proof of the fundamental theorem of algebra in 1799 contained an essentially topological argument; fifty years later, he further developed the topological argument in his fourth proof of this theorem.

Another encounter with topological notions occurred to him in the course of his astronomical work in 1804, when he determined the limits of the region on the celestial sphere in which comets and asteroids might appear, and which he termed "Zodiacus". He discovered that if the Earth's and comet's orbits are linked, then by topological reasons the Zodiacus is the entire sphere. In 1848, in the context of the discovery of the asteroid 7 Iris, he published a further qualitative discussion of the Zodiacus.

From Gauss's letters during the period of 1820–1830, one can learn that he thought intensively on topics with close affinity to Geometria Situs, and became gradually conscious of semantic difficulty in this field. Fragments from this period reveal that he tried to classify "tract figures", which are closed plane curves with a finite number of transverse self-intersections, that may also be planar projections of knots. To do so he devised a symbolical scheme, the Gauss code, that in a sense captured the characteristic features of tract figures.

In a fragment from 1833, Gauss defined the linking number of two space curves by a certain double integral, and in doing so provided for the first time an analytical formulation of a topological phenomenon. On the same note, he lamented the little progress made in Geometria Situs, and remarked that one of its central problems will be "to count the intertwinings of two closed or infinite curves". His notebooks from that period reveal that he was also thinking about other topological objects such as braids and tangles.

Gauss's influence in later years to the emerging field of topology, which he held in high esteem, was through occasional remarks and oral communications to Mobius and Listing.

Minor mathematical accomplishments
Gauss applied the concept of complex numbers to solve some well-known problems in a new concise way. For example in a short note from 1836 on geometric aspects of the ternary forms and their application to crystallography, he stated the fundamental theorem of axonometry, which tells how to represent a 3D cube on a 2D plane with complete accuracy, via complex numbers. He described rotations of this sphere as the action of certain linear fractional transformations on the extended complex plane, and gave a proof for the geometric theorem that the altitudes of a triangle always meet in a single orthocenter.

Gauss was concerned with John Napier's "Pentagramma mirificum" – a certain spherical pentagram – for several decades; he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects. In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane.

Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given quadrilateral, and discovered a surprising result about the computation of area of pentagons.

Geomagnetism
Gauss has been interested in magnetism since 1803. After Alexander von Humboldt visited Göttingen in 1826, both scientists began intensive research on geomagnetism, partly independently, partly in productive cooperation. In 1828, Gauss was Humboldt's guest during the conference of the Society of German Natural Scientists and Physicians in Berlin, where he got acquainted with the physicist Wilhelm Weber.

When Weber got the chair for physics in Göttingen as successor of Johann Tobias Mayer by Gauss's recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge of magnetism with a representation for the unit of magnetism in terms of mass, charge, and time. They founded the Magnetic Association (German: Magnetischer Verein), an international working group of several observatories, which supported measurements of Earth's magnetic field in many regions of the world with equal methods at arranged dates in the years 1836 to 1841.

In 1836, Humboldt suggested the establishment of a worldwide net of geomagnetic stations in the British dominions with a letter to the Duke of Sussex, then president of the Royal Society; the magnetic measures should be taken under standardized conditions using Gauss's methods. Together with other instigators, this led to a global program known as "Magnetical crusade" under directory of Edward Sabine. The dates, times, and intervals of observations were determined in advance, the Göttingen mean time was used as standard. 61 stations on all five continents participated in this global program. Gauss and Weber founded a series for publication of the results, six volumes were edited between 1837 and 1843. Weber's departure to Leipzig in 1843 as late effect of the Göttingen Seven affair marked the end of Magnetic Association activity.

Following Humboldt's example, Gauss ordered a magnetic observatory to be built in the garden of the observatory, but both scientists differed over instrumental equipment; Gauss preferred stationary instruments, which he thought to give more precise results, whereas Humboldt was accustomed to movable instruments. Gauss was interested in the temporal and spatial variation of magnetic declination, inclination, and intensity, but discriminated Humboldt's concept of magnetic intensity to the terms of "horizontal" and "vertical" intensity. Together with Weber, he developed methods of measuring the components of intensity of the magnetic field, and constructed a suitable magnetometer to measure absolute values of the strength of the Earth's magnetic field, not more relative ones that depended on the apparatus. The precision of the magnetometer was about ten times higher than of previous instruments. With this work, Gauss was the first to derive a non-mechanical quantity by basic mechanical quantities.

Gauss carried out a General Theory of Terrestrial Magnetism (1839), in what he believed to describe the nature of magnetic force; according to Felix Klein, this work is a presentation of observations by use of spherical harmonics rather than a physical theory. The theory predicted the existence of exactly two magnetic poles on the Earth, thus Hansteen's idea of four magnetic poles became obsolete, and the data allowed to determine their location with rather good precision.

Gauss influenced the beginning of geophysics in Russia, when Adolph Theodor Kupffer, one of his former students, founded a magnetic observatory in St. Petersburg, following the example of the observatory in Göttingen, and similarly, Ivan Simonov in Kazan.

Electromagnetism
The discoveries of Hans Christian Ørsted on electromagnetism and Michael Faraday on electromagnetic induction drew Gauss's attention to these matters. Gauss and Weber found rules for branched electric circuits, which were later found independently and firstly published by Gustav Kirchhoff and benamed after him as Kirchhoff's circuit laws, and made inquiries on electromagnetism. They constructed the first electromechanical telegraph in 1833, and Weber himself connected the observatory with the institute for physics in the town centre of Göttingen, but they did not care for any further development of this invention for commercial purposes.

Gauss's main theoretical interests in electromagnetism were reflected in his attempts to formulate quantitive laws governing electromagnetic induction. In notebooks from these years, he recorded several innovative formulations; he discovered the idea of vector potential function (independently rediscovered by Franz Ernst Neumann in 1845), and in January 1835 he wrote down an "induction law" equivalent to Faraday's law, which stated that the electromotive force at a given point in space is equal to the instantaneous rate of change (with respect to time) of this function.

Gauss tried to find a unifying law for long-distance effects of electrostatics, electrodynamics, electromagnetism, and induction, comparable to Newton's law of gravitation, but his attempt ended in a "tragic failure".

Potential theory
Since Isaac Newton had shown theoretically that the Earth and rotating stars assume non-spherical shapes, the problem of attraction of ellipsoids gained importance in mathematical astronomy. In his first publication on potential theory, the "Theoria attractionis..." (1813), Gauss provided a closed-form expression to the gravitational attraction of a homogeneous triaxial ellipsoid at every point in space. In contrast to pevious research of Maclaurin, Laplace and Lagrange, Gauss's new solution treated the attraction more directly in the form of an elliptic integral. In the process, he also proved and applied some special cases of the so-called Gauss's theorem in vector analysis.

In the General theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances (1840) Gauss gave the baseline of a theory of the magnetic potential, based on Lagrange, Laplace, and Poisson; it seems rather unlikely that he knew the previous works of George Green on this subject. However, Gauss could never give any reasons for magnetism, nor a theory of magnetism similar to Newton's work on gravitation, that enabled scientists to predict geomagnetic effects in the future.

Optics
Gauss's calculations enabled instrument maker Johann Georg Repsold in Hamburg to construct a new achromatic lens system in 1810. A main problem, among other difficulties, was the nonprecise knowledge of the refractive index and dispersion of the used glass types. In a short article from 1817 Gauss dealt with the problem of removal of chromatic aberration in double lenses, and computed adjustments of the shape and coefficients of refraction required to minimize it. His work was noted by the optician Carl August von Steinheil, who in 1860 introduced the achromatic Steinheil doublet, partly based on Gauss's calculations. Many results in geometrical optics are only scattered in Gauss's correspondences and hand notes. In the Dioptrical Investigations (1840), Gauss gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics). He characterized optical systems under a paraxial approximation only by its cardinal points, and he derived the Gaussian lens formula, applicable without restrictions in respect to the thickness of the lenses.

Mechanics
Gauss's first business in mechanics concerned the earth's rotation. When his university friend Benzenberg carried out experiments to determine the deviation of falling masses from the perpendicular in 1802, what today is known as an effect of the Coriolis force, he asked Gauss for a theory based calculation of the values for comparison with the experimental ones. Gauss elaborated a system of fundamental equations for the motion, and the results corresponded sufficiently with Benzenberg's data, who added Gauss's considerations as an appendix to his book on falling experiments.

After Foucault had demonstrated the earth's rotation by his pendulum experiment in public in 1851, Gerling questioned Gauss for further explanations. This instigated Gauss to design a new apparatus for demonstration with a much shorter length of pendulum than Foucault's one. The oscillations were observed with a reading telescope, with a vertical scale and a mirror fastened at the pendulum. It is described in the Gauss–Gerling correspondence, and Weber made some experiments with this apparatus in 1853, but no data were published.

Gauss's principle of least constraint of 1829 was established as a general concept to overcome the division of mechanics into statics and dynamics, combining D'Alembert's principle with Lagrange's principle of virtual work, and showing analogies to the method of least squares.

Metrology
In 1828, Gauss was appointed to head of a Board for weights and measures of the Kingdom of Hanover. He provided the creation of standards of length and measures. Gauss himself took care of the time-consuming measures and gave detailed orders for the mechanical preparation. In the correspondence with Schumacher, who was also working on this matter, he described new ideas for scales of high precision. He submitted the final reports on the Hanoverian foot and pound to the government in 1841. This work got more than regional importance by the order of a law of 1836, that connected the Hanoverian measures with the English ones.

Anecdotes
Several stories of his early genius have been reported. Carl Friedrich Gauss's mother had never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension, which occurs 39 days after Easter. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years. Gauss felt sorry for his newborn daughter Wilhelmine, because she was born on the leap day in 1808 and thus would celebrate her birthday only every four years.

In his memorial on Gauss, Wolfgang Sartorius von Waltershausen tells a story about the three-year-old Gauss, who corrected a math error his father made. The most popular story, also told by Sartorius, tells of a school exercise: the teacher Büttner and his assistant Martin Bartels ordered students to add an arithmetic series. Out of about a hundred pupils, Gauss was the first to solve the problem correctly by a significant margin. Although (or because) Sartorius gave no details, over time many versions of this story have been created, with more and more details regarding the nature of the series – the most frequent being the classical problem of adding together all the integers from 1 to 100 – and the circumstances in the classroom.

Honours and awards
The first membership of a scientific society was given to Gauss in 1802 by the Russian Academy of Sciences. Further memberships (corresponding, foreign or full) were awarded from the Academy of Sciences in Göttingen (1802/ 1807), the French Academy of Sciences (1804/ 1820), the Royal Society of London (1804), the Royal Prussian Academy in Berlin (1810), the National Academy of Science in Verona (1810), the Royal Society of Edinburgh (1820), the Bavarian Academy of Sciences of Munich (1820), the Royal Danish Academy in Copenhagen (1821), the Royal Astronomical Society in London (1821), the Royal Swedish Academy of Sciences (1821), the American Academy of Arts and Sciences in Boston (1822), the Royal Bohemian Society of Sciences in Prague (1833), the Royal Academy of Science, Letters and Fine Arts of Belgium (1841/ 1845), the Royal Society of Sciences in Uppsala (1843), the Royal Irish Academy in Dublin (1843), the Royal Institute of the Netherlands (1845/ 1851), the Spanish Royal Academy of Sciences in Madrid (1850), the Russian Geographical Society (1851), the Imperial Academy of Sciences in Vienna (1848), the American Philosophical Society (1853), the Cambridge Philosophical Society, and the Royal Hollandish Society of Sciences in Haarlem.

Both the University of Kazan and the Philosophy Faculty of the University of Prague appointed him honorary member in 1848.

Gauss received the Lalande Prize from the French Academy of Science in 1809 for the theory of planets and the means of determining their orbits from only three observations, the Danish Academy of Science prize in 1823 for his memoir on conformal projection, and the Copley Medal from the Royal Society in 1838 for "his inventions and mathematical researches in magnetism".

Gauss was appointed Knight of the French Legion of Honour in 1837, and was taken as one of the first members of the Prussian Order Pour le Merite (Civil class) when it was established in 1842. He received the Order of the Crown of Westphalia (1810), the Danish Order of the Dannebrog (1817), the Hanoverian Royal Guelphic Order (1815), the Swedish Order of the Polar Star (1844), the Order of Henry the Lion (1849), and the Bavarian Maximilian Order for Science and Art (1853).

The Kings of Hanover appointed him the honorary titles "Hofrath" (1816) and "Geheimer Hofrath" (1845). In 1949, on the occasion of his golden doctor degree jubilee, he got the honorary citizenship of both towns of Brunswick and Göttingen. Soon after his death a medal was issued by order of King George V of Hanover with the back side inscription dedicated "to the Prince of Mathematicians".

The "Gauss-Gesellschaft Göttingen" ("Göttingen Gauss Society") was founded in 1964 for research on life and work of Carl Friedrich Gauss and related persons and edits the Mitteilungen der Gauss-Gesellschaft (Communications of the Gauss Society).

Names and commemorations

 * List of things named after Carl Friedrich Gauss

Mathematics and astronomy

 * 1799: (Doctoral thesis on the fundamental theorem of algebra, University of Helmstedt) Original book
 * 1816: Original
 * 1816: Original
 * 1850:    Original (Lecture from 1849)
 * (German)
 * 1800: Original
 * 1801:
 * (translated from the second German edition, Göttingen 1860)
 * 1802: Original
 * 1804: Original (on the Zodiacus)
 * 1808: Original (Introduces Gauss's lemma, uses it in the third proof of quadratic reciprocity)
 * 1808:
 * 1809: Original book
 * 1811: Original (from 1810) (Orbit of Pallas)
 * 1811:  Original (from 1808) (Determination of the sign of the quadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity)
 * 1813: Original (from 1812, contains the Gauss's continued fraction)
 * 1816: Original (from 1814)
 * 1818: Original (from 1817) (Fifth and sixth proofs of quadratic reciprocity)
 * 1818: Original (Only reference to the – mostly unpublished – work on the algorithm of the arithmetic-geometric mean.)
 * 1823:  Original (from 1821)
 * 1823: Original
 * 1825: (Prize winning essay from 1822 on conformal mapping)
 * 1828: Original book
 * 1828: (from 1826)
 * (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation)
 * 1828:    Original (from 1827)
 * 1828: Original (from 1825)
 * 1832: Original (from 1831) (Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + i)
 * 1845: Original (from 1843)
 * 1847: Original (from 1846)
 * 1848: Original
 * 1903: Wissenschaftliches Tagebuch  Original book (from 1847, on the Zodiacus)
 * 1848: Original
 * 1903: Wissenschaftliches Tagebuch  Original book (from 1847, on the Zodiacus)

Physics

 * 1804: Fundamentalgleichungen für die Bewegung schwerer Körper auf der Erde ( in original book: Original)
 * 1813: Original (contains Gauss's theorem of vector analysis)
 * 1817:
 * 1829:
 * 1830: Original (from 1829)
 * 1841:    Original (from 1832)
 * The Intensity of the Earth's Magnetic Force Reduced to Absolute Measurement. Translated by Susan P. Johnson.
 * 1836: Erdmagnetismus und Magnetometer (Original book: )
 * 1840: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte (Original book:
 * 1843: Original (from 1840)

Together with Wilhelm Weber

 * 1837–1839:
 * 1840–1843:
 * 1840:

Collected works

 * (includes unpublished literary estate)

Correspondence

 * (letters from December 1804 to August 1844)
 * (letters to Boguslawski from February 1835 to January 1848)
 * (letters from February 1802 to October 1826)
 * (letters from September 1797 to February 1853; added letters of other correspondents)
 * (letters from June 1810 to June 1854)
 * (letters from June 1810 to June 1854)
 * (letters from July 1807 to December 1854; added letters of other correspondents)
 * (letters from 1835 to 1843)
 * (letters from February 1799 to September 1800)
 * (letters from January 1802 to October 1819)
 * (letters from January 1820 to May 1839; added letters of other correspondents)
 * Volumes 1+2 (letters from April 1808 to March 1836)
 * Volumes 3+4 (letters from March 1836 to April 1845)
 * Volumes 5+6 (letters from April 1845 to November 1850)
 * (letters from 1795 to 1815)
 * Volumes 3+4 (letters from March 1836 to April 1845)
 * Volumes 5+6 (letters from April 1845 to November 1850)
 * (letters from 1795 to 1815)

The Göttingen Academy of Sciences and Humanities provides a complete collection of the yet known letters from and to Carl Friedrich Gauss that is accessible online. The literary estate is kept and provided by the Göttingen State and University Library. Written estate from Carl Friedrich Gauss and family members can also be found in the municipal archive of Brunswick.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Carl Friedrich Gauss


text:
Talk:Charles Darwin

Small fix on the lead
Charles Robert Darwin FRS FRGS FLS FZS JP (/ˈdɑːrwɪn/ DAR-win; 12 February 1809 – 19 April 1882) was an English naturalist,geologist and biologist ...

There should be a space: naturalist, geologist ... 2804:14D:BE81:84F2:E081:8EEA:3208:A16F (talk) 23:54, 4 November 2023 (UTC)

 * ✅: well spotted, thanks! AntiDionysius (talk) 23:57, 4 November 2023 (UTC)

Lede image
Wouldn't it be better to use this Featured image of Darwin instead of the current one?

GuardianH (talk) 04:40, 14 February 2024 (UTC)
 * The current article is fine. It includes the above image with a suitable caption (search for "between 1862 and 1866"). Johnuniq (talk) 05:14, 14 February 2024 (UTC)

1860 debate
The account here is the popular one, but it is probably apocryphal. Wikipedia has a page on this debate which gives a more accurate version. MirrorSquirrel (talk) 18:32, 6 June 2024 (UTC)

Darwin's support for the inheritance of acquired characters
It should be made clear that in "The Origin" Darwin attributes several adaptions to what he calls "use and disuse" which involves the inheritance of acquired characteristics which is a component of Lamarck's theory of evolution. Currently the article mentions that three of Darwin's contemporaries (Robert Grant, Ernst Haeckel and Herbert Spencer) held Lamarckian views. It also describes one of Darwin's notes as dismissing as absurd the Lamarckian view of progression to higher forms. This gives the impression that Darwin had discarded Larmarckian thinking which is wrong. Darwin argues for Lamarckian evolutionary mechanisms in "The Origin" and even more so in the later editions and other subsequent published material. References:

"Darwinian evolution includes Lamarckian inheritance of acquired characters", International Journal of Epidemiology, Yongsheng Liu, Volume 45, Issue 6, December 2016, Pages 2206–2207 (https://doi.org/10.1093/ije/dyw182) [This paper notes the (apparently) growing support for the inheritance of acquired characters today and suggests that Darwin was actually correct in his adherence to it]

"Lamarckism" in Encyclopedia Britannica (https://www.britannica.com/science/Lamarckism) [See the section titled "The influence of Lamarckism"]

It should also be stated in the article that Darwin did not understand that natural selection was responsible for species conformity (stabilizing selection) and this led him to articulate, in "The Origin" a confusing theory about variation being dependent on a change to the "conditions of life" (see the second sentence of "The Origin").

Reference: "Darwin on Variation and Heredity", Rasmus G. Winther, Journal of the History of Biology 33: 425–455, 2000 (https://philpapers.org/archive/WINDOV)

"The Development of Darwin's Theory" (book) by Dov Ospovat, Cambridge University Press [pages 77-83 (paperback edition) talk about Darwin's ideas on variation].

These two intimately related issues are interesting and noteworthy and, by including them in the article, will give a fuller, more balanced and more interesting account of Darwin. Failing to appreciate these points makes reading the Origin of Species a confusing exercise. — Preceding unsigned comment added by Gourdiehill (talk • contribs) 23:05, 22 June 2024 (UTC)
 * Do you have any thoughts on this? To me, it seems a bit unnecessary to mention that 160-year old initial investigations concerning a ground-breaking theory might have had some rough edges. Is there something in the current article that is misleading? Is there a reason the reference in the article concerning the issue should be doubted? Johnuniq (talk) 04:02, 24 June 2024 (UTC)
 * The meaning of "Lamarckism" changed after Darwin's time, see On the Origin of Species: Charles Darwin's grandfather Erasmus Darwin outlined a hypothesis of transmutation of species in the 1790s, and French naturalist Jean-Baptiste Lamarck published a more developed theory in 1809. Both envisaged that spontaneous generation produced simple forms of life that progressively developed greater complexity, adapting to the environment by inheriting changes in adults caused by use or disuse. This process was later called Lamarckism. Lamarck thought there was an inherent progressive tendency driving organisms continuously towards greater complexity, in parallel but separate lineages with no extinction. .... Some anatomists such as Robert Grant were influenced by Lamarck and Geoffroy. This article rightly describes one of Darwin's notes as dismissing as absurd the Lamarckian view of progression to higher forms – Darwin discarded that central Larmarckian idea. Lamarckism notes that CD proposed natural selection as the main mechanism for development of species, but (like Lamarck) gave credence to the idea of heritable effects of use and disuse as a supplementary mechanism. There's a case for a very brief mention of this, but the focus has to remain on concisely outlining Darwin's theory of natural selection. . . dave souza, talk 06:43, 24 June 2024 (UTC)

title: Talk:Charles Darwin