The quadrature of the hyperbola was first accomplished by Gregoire de Saint-Vincent in 1647 in his momentous Opus geometricum quadrature circuli et sectionum coni.

: made the quadrature of a hyperbola to its asymptotes, and showed that as the area increased in arithmetic series the abscissas increased in geometric series.

His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola, which requires approximation of the natural logarithm function by infinite series.

The quadrature of the hyperbola by Saint-Vincent and de Sarasa provided a new function, the natural logarithm, of critical importance.

To a large extent, recognition of de Saint-Vincent's achievement in quadrature of the hyperbola is due to his student and co-worker Alphonse Antonio de Sarasa, with Marin Mersenne acting as catalyst.

Aberration is usually larger than a planet's light-time correction except when the planet is near quadrature ( 90 ° from the Sun ), where aberration drops to zero because then the Earth is directly approaching or receding from the planet.

The Euler – Maclaurin formula is also used for detailed error analysis in numerical quadrature.

Clenshaw – Curtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler – Maclaurin approach is very accurate ( in that particular case the Euler – Maclaurin formula takes the form of a discrete cosine transform ).

Assume the Earth is in L, at the second quadrature with Jupiter ( i. e. ALB is 90 °), and Io emerges from D. After several orbits of Io, at 42. 5 hours per orbit, the Earth is in K. Rømer reasoned that if light is not propagated instantaneously, the additional time it takes to reach K, that he reckoned about 3½ minutes, would explain the observed delay.

The other component, which is always 90 ° ( radians ) " out of phase ", is referred to as the quadrature component.

Special-purpose I / O modules, such as timer modules or counter modules, can be used where the scan time of the processor is too long to reliably pick up, for example, counting pulses and interpreting quadrature from a shaft encoder.

Single-sideband has the mathematical form of quadrature amplitude modulation in the special case where one of the baseband waveforms is derived from the other, instead of being independent messages:

Another variation, the Weaver modulator, uses only lowpass filters and quadrature mixers, and is a favored method in digital implementations.

In Weaver's method, the band of interest is first translated to be centered at zero, conceptually by modulating a complex exponential with frequency in the middle of the voiceband, but implemented by a quadrature pair of sine and cosine modulators at that frequency ( e. g. 2 kHz ).

Then, the single-sideband complex signal centered at zero is upconverted to a real signal, by another pair of quadrature mixers, to the desired center frequency.

The equivalent baseband signal is where is the inphase signal, the quadrature phase signal, and the imaginary unit.

It is this quadrature phase relationship which creates the helix and causes the points of maximum magnitude of the vertical component to correspond with the points of zero magnitude of the horizontal component, and vice versa.

Also, if the two dipoles were fed with a 90 ° degree time-phase difference ( phase quadrature ), the polarization along zenith would be circular ... One way to obtain the 90 ° time-phase difference between the two orthogonal field components, radiated respectively by the two dipoles, is by feeding one of the two dipoles with a transmission line which is 1 / 4 wavelength longer or shorter than that of the other ", p. 80 ;

The quadrature phase difference is obtained by feeding the element with a 90 ° power divider ", p. 859.

The heterodyne system in these cases is used to convert quadrature phase-encoded and amplitude modulated sine waves from the broadcast frequencies to frequencies recordable in less than 1 MHz bandwidth.

In the near field is a quadrature component of the particle velocity that is 90 ° out of phase with the sound pressure and does not contribute to the time-averaged energy or the intensity of the sound.

The accuracy of a quadrature rule of the Newton-Cotes type is generally a function of the number of evaluation points.

The most famous of these problems, squaring the circle, otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass.

Historically the power rule was derived as the inverse of Cavalieri's quadrature formula which gave the area under for any integer.

The term quadrature of the circle is sometimes used synonymously, or may refer to approximate or numerical methods for finding the area of a circle.

The problem of finding the area under a curve, known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus.

The area of the surface included between this curve and its base is found – the first known instance of a quadrature of a curved surface.

be used to accomplish quadrature of a rectangle ( since a square whose sides are equal to the geometric mean of the sides of a rectangle has the same area as the rectangle ), and thus of any figure for which we can construct a rectangle of equal area, such as any polygon ( but not a circle ).

The idea behind this is that the circle denoting a coherent state in a quadrature diagram ( see below ) has been " squeezed " to an ellipse of the same area.

On the integral side, Cavalieri developed his method of indivisibles in the 1630s and 40s, providing a more modern form of the ancient Greek method of exhaustion, and computing Cavalieri's quadrature formula, the area under the curves x < sup > n </ sup > of higher degree, which had previously only been computed for the parabola, by Archimedes.

These make analytic integrals often easier to evaluate, and Gaussian quadrature tables are often presented in terms of area coordinates.

In medieval Europe the quadrature meant calculation of area by any method.

The two carrier waves, usually sinusoids, are out of phase with each other by 90 ° and are thus called quadrature carriers or quadrature components — hence the name of the scheme.

* Quadrature amplitude modulation ( QAM ), a modulation method of using both an ( in-phase ) carrier wave and a ' quadrature ' carrier wave that is 90 ° out of phase with the main, or in-phase, carrier

One may consider this clock pulse redundant information, or at least a wasteful use of channel capacity, and duplex the channel by varying the phase, as in polar modulation, or adding another signal that is 90 ° out of phase ( a sine wave ), as in quadrature modulation.

A stereo phaser is usually two identical phasers modulated by a quadrature signal ; the output of the oscillators for the left and right channels are a quarter-wave out of phase.

" He wrote two books on the subject, one published in 1647 and the other in 1668, which cover some two or three thousand closely printed pages ; the fallacy in the quadrature was pointed out by Christiaan Huygens.

At this time the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of x < sup > n </ sup > up to degree in Cavalieri's quadrature formula.

* Quadrature phase, oscillations that are said to be in quadrature if they are separated in phase by 90 ° ( π / 2, or λ / 4 )

An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points x < sub > i </ sub > and weights w < sub > i </ sub > for i = 1 ,..., n.

Gaussian quadrature as above will only produce accurate results if the function f ( x ) is well approximated by a polynomial function within the range.

DCTs are also closely related to Chebyshev polynomials, and fast DCT algorithms ( below ) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in Clenshaw – Curtis quadrature.

* The quadrature detector, which phase shifts the signal by 90 degrees and multiplies it with the unshifted version.

This integral is called Cavalieri's quadrature formula and was first found in a geometric form by Bonaventura Cavalieri for.

A large class of quadrature rules can be derived by constructing interpolating functions which are easy to integrate.

Heuristics for adaptive quadrature are discussed by Forsythe et al.

) Hence, if we approximate the integral ∫< sub > a </ sub >< sup > b </ sup > f ( x ) dx by the quadrature rule ( b − a ) f ( a ) our error is no greater than the right hand side of (**).

) Using more derivatives, and by tweaking the quadrature, we can do a similar error analysis using a Taylor series ( using a partial sum with remainder term ) for f. This error analysis gives a strict upper bound on the error, if the derivatives of f are available.

Sparse grids were originally developed by Smolyak for the quadrature of high dimensional functions.

Hence he inferred that the quadrature of the circle was impossible ; this was accepted by Montucla, but it is not conclusive, for it is conceivable that some particular sector might be squared, and this particular sector might be the whole circle.

Similar to MP3, ATRAC uses stacked quadrature mirror filters ( QMF ) followed by an MDCT.

NCOs are phase-and frequency-agile, and can be trivially modified to produce phase-modulated or frequency-modulated by summation at the appropriate node, or provide quadrature outputs as shown in the figure.

This include various trigonometric expansions, trigonometric approximations and methods which take advantage of the quadrature symmetry exhibited by sinusoids.