* 1777 – Carl Friedrich Gauss, German mathematician ( d. 1855 )

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

* Gauss – Kronrod quadrature formula

The most well-known use of the Cooley – Tukey algorithm is to divide the transform into two pieces of size at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general ( as was known to both Gauss and Cooley / Tukey ).

The two monographs Gauss published on biquadratic reciprocity have consecutively-numbered sections: the first contains §§ 1 – 23 and the second §§ 24 – 76.

The Gauss – Legendre algorithm is an algorithm to compute the digits of π.

The method is based on the individual work of Carl Friedrich Gauss ( 1777 – 1855 ) and Adrien-Marie Legendre ( 1752 – 1833 ) combined with modern algorithms for multiplication and square roots.

Gauss – Jordan elimination, an extension of this algorithm, reduces the matrix further to diagonal form, which is also known as reduced row echelon form.

Gaussian elimination alone is sufficient for many applications, and requires fewer calculations than the Gauss – Jordan version.

At the end of the algorithm, if the Gauss – Jordan elimination ( zeros under and above the leading 1 ) is applied:

By the 1830s mathematics, physics, chemistry, and biology had emerged with world class science, led by Alexander von Humboldt ( 1769 – 1859 ) in natural science and Carl Friedrich Gauss ( 1777 – 1855 ) in mathematics.

File: Carl Friedrich Gauss. jpg | Carl Friedrich Gauss ( 1777 – 1855 )

Other standard iterative methods for matrix equation solutions can also be used, for example the Gauss – Seidel method, where updated values for each patch are used in the calculation as soon as they are computed, rather than all being updated synchronously at the end of each sweep.

* Gauss – Lucas theorem

There are two versions of the first message sent by Gauss and Weber: the more official one is based on a note in Gauss's own handwriting stating that " Wissen vor meinen – Sein vor scheinen " (" knowing before opining, being before seeming ") was the first message sent over the electromagnetic telegraph.

The Gauss – Markov theorem states that the estimate of the mean having minimum variance is given by:

However, in the pure Gauss – Bonnet gravity ( a modification to general relativity involving extra spatial dimensions which is sometimes studied in the context of brane cosmology ) exotic matter is not needed in order for wormholes to exist — they can exist even with no matter.

* May 6 – Carl Friedrich Gauss and Wilhelm Weber obtain permission to build an electromagnetic telegraph in Göttingen.

* March 30 – Carl Gauss obtains conditions for the constructibility by ruler and compass of regular polygons, and is able to announce that the regular 17-gon is constructible by ruler and compasses.

* July 10 – Carl Friedrich Gauss discovers that every positive integer is representable as a sum of at most 3 triangular numbers.

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic ; see the Gauss – Bonnet theorem.

The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the ( angular ) defect ; the analog for the Gauss – Bonnet theorem is Descartes ' theorem on total angular defect.

It is named after Carl Friedrich Gauss who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848.

Consider for instance the open unit disc, a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss – Bonnet formula does not work.

Then we can apply Gauss – Bonnet to the surface T formed by the inside of that triangle and the piecewise boundary given by the triangle itself.

A number of earlier results in spherical geometry and hyperbolic geometry over the preceding centuries were subsumed as special cases of Gauss – Bonnet.

This is the special case of Gauss – Bonnet, where the curvature is concentrated at discrete points ( the vertices ).

Thinking of curvature as a measure, rather than as a function, Descartes ' theorem is Gauss – Bonnet where the curvature is a discrete measure, and Gauss – Bonnet for measures generalizes both Gauss – Bonnet for smooth manifolds and Descartes ' theorem.

There are several combinatorial analogs of the Gauss – Bonnet theorem.

Generalizations of the Gauss – Bonnet theorem to n-dimensional Riemannian manifolds were found in the 1940s, by Allendoerfer, Weil, and Chern ; see generalized Gauss – Bonnet theorem and Chern – Weil homomorphism.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, 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.

The theorem was conjectured by Euler and Legendre and first proven by Gauss.

She asked Gauss if her approach to the theorem was worth pursuing.

If we choose the volume to be a ball of radius a around the source point, then Gauss ' divergence theorem implies that

* 1942 – Joseph Leo Doob states his theorem on Gauss – Markov processes

This result is known as the Gauss – Markov theorem.

* The Gauss – Markov theorem.

The Gauss – Markov theorem shows that, when this is so, is a best linear unbiased estimator ( BLUE ).

* Theorema Egregium − The " remarkable theorem " discovered by Gauss which showed there is an intrinsic notion of curvature for surfaces.

Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula.

The formula was described by Meister in 1769 and by Gauss in 1795.

* Carl Friedrich Gauss pioneers the field of summation with the formula summing 1: n as ( n ( n + 1 ))/ 2, at the age of 7.

For example, it follows that any closed oriented Riemannian surface can be C < sup > 1 </ sup > isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space ( there is no such C < sup > 2 </ sup >- embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε < sup >- 2 </ sup >).

In 1830, Carl Friedrich Gauss, the German mathematician, unified the work of these two scientists to derive the Young – Laplace equation, the formula that describes the capillary pressure difference sustained across the interface between two static fluids.

Gauss ' constant can be used as the constant of proportionality in the formula for the mean daily motion, n ( in radians per day ), for bodies in elliptical orbits.

* Carl Friedrich Gauss, at the age of seven, pioneers the field of summation with the formula summing 1: n as ( n ( n + 1 ))/ 2.

Here is the Legendre symbol, which is a quadratic character mod p. An analogous formula with a general character χ in place of the Legendre symbol defines the Gauss sum G ( χ ).

* The exact value of the Gauss sum, computed by Gauss, is given by the formula

In his second monograph on biquadratic reciprocity, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of 1 + i in Z, the ring of Gaussian integers.