Gauss's method involved determining a conic section in space, given one focus ( the Sun ) and the conic's intersection with three given lines ( lines of sight from the Earth, which is itself moving on an ellipse, to the planet ) and given the time it takes the planet to traverse the arcs determined by these lines ( from which the lengths of the arcs can be calculated by Kepler's Second Law ).

The equations are then discretized onto a grid using the finite difference method and solved subject to the constraints of Gauss's law and Linear elasticity.

An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid Ceres.

* The squared-error loss function is widely used in statistics, following Gauss's use of the squared-error loss function in justifying the method of least squares.

Bertrand translated into French Carl Friedrich Gauss's work on the theory of errors and the method of least squares.

In one case he reportedly gave a method equivalent to Gauss's pivotal condensation.

These orbits were stabilized in the model by the fact that when an electron moved farther from the center of the positive cloud, it felt a larger net positive inward force, because there was more material of opposite charge, inside its orbit ( see Gauss's law ).

Two individuals gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer.

According to Dunnington, Gauss's religion was based upon the search for truth.

Gauss's personal life was overshadowed by the early death of his first wife, Johanna Osthoff, in 1809, soon followed by the death of one child, Louis.

Of all of Gauss's children, Wilhelmina was said to have come closest to his talent, but she died young.

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.

Gauss's story was well known in Doyle's time, and Ramanujan's story unfolded at Cambridge from early 1913 to mid 1914 ; The Valley of Fear, which contains the comment about maths so abstruse that no one could criticise it, was published in September 1914.

Bernhard Riemann extended Gauss's theory to higher dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces.

It was concluded that magnetic monopoles did not exist: One of Maxwell's equations, now called Gauss's law for magnetism, is the mathematical statement that there are no magnetic monopoles.

Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture.

Although Kummer had studied Fermat's Last Theorem in the 1830s and was probably aware that his theory would have implications for its study, it is more likely that the subject of Jacobi's ( and Gauss's ) interest, higher reciprocity laws, held more importance for him.

: The fact that was easy to prove and led to one of Gauss's proofs of quadratic reciprocity.

, one of Gauss's last students and a historian of mathematics, who was summarizing a remark made by Gauss about Eisenstein in a conversation many years earlier.

Gauss's law allows the E-field to be calculated in terms of a continuous distribution of charge density

Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions, and secondly on his proof that this can be done if the odd prime factors of n are distinct Fermat primes, which are of the form.

This can also be expressed in terms of vector field quantities by taking the divergence of Ampère's law with Maxwell's correction and combining with Gauss's law, yielding:

Similarly, the total mass inside a sphere containing a black hole can be found by using the gravitational analog of Gauss's law, the ADM mass, far away from the black hole.

Gauss's intellectual abilities attracted the attention of the Duke of Braunschweig, who sent him to the Collegium Carolinum ( now Technische Universität Braunschweig ), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798.

) Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges.

Gauss's law describes the relationship between an electric field and the electric charges that cause it: The electric field points away from positive charges and towards negative charges.

The absence of net charge and momentum would follow from accepted physical laws ( Gauss's law and the non-divergence of the stress-energy-momentum pseudotensor, respectively ), if the universe were finite.

The conversion factor can be derived from Gauss's law:

The magnetic field B as calculated from the Biot – Savart law will always satisfy Ampère's circuital law and Gauss's law for magnetism.

Magnetic field lines only exist as loops, they cannot diverge from or converge to a point like electric field lines can ( see Gauss's law for magnetism ).

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German.

In electromagnetism one can derive the energy density of a field from Gauss's law, assuming the curl of the field is 0.

It follows from an application of Gauss's Lemma that if A is the norm of then the distance, induced by the metric, between two close enough points on the curve γ, say γ ( t < sub > 1 </ sub >) and γ ( t < sub > 2 </ sub >), is given by

Gauss's Theorema Egregium ( Latin: " remarkable theorem ") states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself.

* Gauss's inequality, a similar result for the distance from the mode rather than the mean

When it came to deriving the electromagnetic wave equation from displacement current in his 1865 paper A Dynamical Theory of the Electromagnetic Field, he got around the problem of the non-zero divergence associated with Gauss's law and dielectric displacement by eliminating the Gauss term and deriving the wave equation exclusively for the solenoidal magnetic field vector.

Some theorems from Gauss's theory of circle division

But now, even with 1 / ε < sub > 0 </ sub > not appearing, it has an implied value of 4π and no form of Gauss's law will avoid that factor from scaling the enclosed charge Q.

Defining the electric and magnetic fields from potentials automatically satisfies two of Maxwell's equations: Gauss's law for magnetism and Faraday's Law.

This follows directly from Gauss's law, by integrating over a small rectangular pillbox straddling one plate of the capacitor:

Gauss's formula alternately adds new points at the left and right ends, thereby keeping the set of points centered near the same place ( near the evaluated point ).