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Butler Act

The Butler Act was a 1925 Tennessee law prohibiting public school teachers from denying the book of Genesis account of mankind's origin. The law also prevented the teaching of the evolution of man from what it referred to as lower orders of animals in place of the Biblical account. The law was introduced by Tennessee House of Representatives member John Washington Butler, from whom the law got its name. It was enacted as Tennessee Code Annotated Title 49 (Education) Section 1922, having been signed into law by Tennessee governor Austin Peay.

The law was challenged later that year in a famous trial in Dayton, Tennessee called the Scopes Trial which included a raucous confrontation between prosecution attorney and fundamentalist religious leader, William Jennings Bryan, and noted defense attorney and religious agnostic, Clarence Darrow. It was repealed in 1967.

Provisions of the law
The law, "An act prohibiting the teaching of the Evolution Theory in all the Universities, and all other public schools of Tennessee, which are supported in whole or in part by the public school funds of the State, and to provide penalties for the violations thereof" (Tenn. HB 185, 1925) specifically provided:
 * That it shall be unlawful for any teacher in any of the Universities, Normals and all other public schools of the State which are supported in whole or in part by the public school funds of the State, to teach any theory that denies the Story of the Divine Creation of man as taught in the Bible, and to teach instead that man has descended from a lower order of animals.

It additionally outlined that an offending teacher would be guilty of a misdemeanor and fined between $100 and $500 for each offense.

By the terms of the statute, it could be argued, it was not illegal to teach evolution in respect to non-human creatures, such as that apes descended from protozoa or to teach the mechanisms of variation and natural selection. The bill also did not touch on, or restrict the teaching of prevailing scientific theories of geology or the age of the Earth. It did not even require that the Genesis story be taught, but prohibited solely the teaching that man evolved, or any other theory denying that man was created by God as recorded in Genesis. However the author of the law, a Tennessee farmer and member of the Tennessee House of Representatives John Washington Butler, specifically intended that it would prohibit the teaching of evolution. He later was reported to have said "No, I didn't know anything about evolution when I introduced it. I'd read in the papers that boys and girls were coming home from school and telling their fathers and mothers that the Bible was all nonsense." After reading copies of William Jennings Bryan's lecture "Is the Bible True?" as well as Charles Darwin's The Origin of Species and The Descent of Man, Butler decided the teaching of evolution was dangerous.

Challenges
The law was challenged by the ACLU in the famed Scopes Trial, in which John Scopes, a high school science teacher who agreed to be paid on a charge of having taught evolution, and was nominally served a warrant on May 5, 1925. Scopes was indicted on May 25 and ultimately convicted; on appeal the Tennessee Supreme Court found the law to be constitutional under the Tennessee State Constitution, because:

"We are not able to see how the prohibition of teaching the theory that man has descended from a lower order of animals gives preference to any religious establishment or mode of worship. So far as we know, there is no religious establishment or organized body that has in its creed or confession of faith any article denying or affirming such a theory. — Scopes v. State 289 S.W. 363, 367 (Tenn. 1927)"

Despite this decision, the Tennessee Supreme Court reversed the conviction on a technicality (that the jury should have fixed the amount of the fine), and the case was not retried. During the trial, Butler told reporters: "I never had any idea my bill would make a fuss. I just thought it would become a law, and that everybody would abide by it and that we wouldn't hear any more of evolution in Tennessee."

The law remained on the books until 1967, when teacher Gary L. Scott of Jacksboro, Tennessee, who had been dismissed for violation of the act, sued for reinstatement, citing his First Amendment right to free speech. Although his termination was rescinded, Scott continued his fight with a class action lawsuit in the Nashville Federal District Court, seeking a permanent injunction against enforcement of that law. Within three days of his filing suit, a bill for repeal of the Butler Act had passed both houses of the Tennessee legislature and was signed into law May 18 by Governor Buford Ellington.

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title: Butler Act


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Ante-Nicene Fathers (book)

The Ante-Nicene Fathers, subtitled "The Writings of the Fathers Down to A.D. 325" and abbreviated as ANF, is a collection of books in 10 volumes (one volume is indexes) containing English translations of the majority of Early Christian writings. The period covers the beginning of Christianity until the promulgation of the Nicene Creed at the First Council of Nicaea.

Publication
The series was originally published between 1867 and 1873 by the Presbyterian publishing house T. & T. Clark in Edinburgh under the title Ante-Nicene Christian Library (ANCL), as a response to the Oxford movement's Library of the Fathers which was perceived as too strongly identified with the Anglo-Catholic movement. The volumes were edited by Rev. Alexander Roberts and James Donaldson. This series was available by subscription, but the editors were unable to interest enough subscribers to commission a translation of the homilies of Origen.

In 1885 the Christian Literature Company, first of Buffalo, then New York, began to issue the volumes in a reorganized form. This was done without permission, and was indeed a pirate edition. The new series was edited by the Episcopal bishop of New York, A. Cleveland Coxe. Coxe gave his series the title The Ante-Nicene Fathers. By 1896, this American edition/revision was complete. Unable to close down the pirate, T. & T. Clark were obliged to make what terms they could.

In 1897, a volume 9, which contained new translations, was published by T. & T. Clark as an additional volume, to complete the original ANCL. Apart from volume 9, the contents entirely derived from the ANCL, but in a more chronological order. Coxe added his own introductions and notes, which were criticized by academic authorities and Roman Catholic reviewers.

T. & T. Clark then associated with the Christian Literature Company and with other American publishers for the publication of the Nicene and Post-Nicene Fathers.

Volumes

 * Volume 1: Apostolic Fathers, Justin Martyr, Irenaeus
 * Volume 2: Hermas, Tatian, Athenagoras, Theophilus, Clement of Alexandria
 * Volume 3: Tertullian
 * Volume 4: Tertullian (IV), Minucius Felix, Commodian, Origen
 * Volume 5: Hippolytus, Cyprian, Caius, Novatian, Appendix
 * Volume 6: Gregory Thaumaturgus, Dionysius the Great, Julius Africanus, Anatolius and Minor Writers, Methodius, Arnobius
 * Volume 7: Lactantius, Venantius, Asterius, Victorinus, Dionysius, Apostolic Teaching and Constitutions, Homily, and Liturgies
 * Volume 8: Twelve Patriarchs, Excerpts and Epistles, The Clementina, Aprocryphal Gospels and Acts, Syriac Documents
 * Volume 9: Gospel of Peter, Diatessaron, Testament of Abraham, Epistles of Clement, Origen and Miscellaneous Works
 * Volume 10: Bibliography, General Index, Annotated Index of Authors and Works

title: Ante-Nicene Fathers (book)


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User:Gustavo Wombat

Yet another wikipediast.

Pages that I have worked on include: Witold Gombrowicz, Charles Mingus

title: User:Gustavo Wombat


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Nyquist–Shannon sampling theorem

The Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample rate must be at least twice the bandwidth of the signal to avoid aliasing. In practice, it is used to select band-limiting filters to keep aliasing below an acceptable amount when an analog signal is sampled or when sample rates are changed within a digital signal processing function.

The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.

Strictly speaking, the theorem only applies to a class of mathematical functions having a Fourier transform that is zero outside of a finite region of frequencies. Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are band-limited to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples.

Perfect reconstruction may still be possible when the sample-rate criterion is not satisfied, provided other constraints on the signal are known (see below and compressed sensing). In some cases (when the sample-rate criterion is not satisfied), utilizing additional constraints allows for approximate reconstructions. The fidelity of these reconstructions can be verified and quantified utilizing Bochner's theorem.

The name Nyquist–Shannon sampling theorem honours Harry Nyquist and Claude Shannon, but the theorem was also previously discovered by E. T. Whittaker (published in 1915), and Shannon cited Whittaker's paper in his work. The theorem is thus also known by the names Whittaker–Shannon sampling theorem, Whittaker–Shannon, and Whittaker–Nyquist–Shannon, and may also be referred to as the cardinal theorem of interpolation.

Introduction
Sampling is a process of converting a signal (for example, a function of continuous time or space) into a sequence of values (a function of discrete time or space). Shannon's version of the theorem states:

$$

A sufficient sample-rate is therefore anything larger than $$2B$$ samples per second. Equivalently, for a given sample rate $$f_s$$, perfect reconstruction is guaranteed possible for a bandlimit $$B < f_s/2$$.

When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing. Modern statements of the theorem are sometimes careful to explicitly state that $$x(t)$$ must contain no sinusoidal component at exactly frequency $$B,$$ or that $$B$$ must be strictly less than one half the sample rate. The threshold $$2B$$ is called the Nyquist rate and is an attribute of the continuous-time input $$x(t)$$ to be sampled. The sample rate must exceed the Nyquist rate for the samples to suffice to represent $$x(t).$$ The threshold $$f_s/2$$ is called the Nyquist frequency and is an attribute of the sampling equipment. All meaningful frequency components of the properly sampled $$x(t)$$ exist below the Nyquist frequency. The condition described by these inequalities is called the Nyquist criterion, or sometimes the Raabe condition. The theorem is also applicable to functions of other domains, such as space, in the case of a digitized image. The only change, in the case of other domains, is the units of measure attributed to $$t,$$ $$f_s,$$ and $$B.$$

The symbol $$T \triangleq 1/f_s$$ is customarily used to represent the interval between samples and is called the sample period or sampling interval. The samples of function $$x(t)$$ are commonly denoted by $$x[n] \triangleq T\cdot x(nT)$$ (alternatively $$x_n$$ in older signal processing literature), for all integer values of $$n.$$  The $$T$$ multiplier is a result of the transition from continuous time to discrete time (see Discrete-time_Fourier_transform), and it preserves the energy of the signal as $$T$$ varies.

A mathematically ideal way to interpolate the sequence involves the use of sinc functions. Each sample in the sequence is replaced by a sinc function, centered on the time axis at the original location of the sample $$nT,$$ with the amplitude of the sinc function scaled to the sample value, $$x(nT).$$ Subsequently, the sinc functions are summed into a continuous function. A mathematically equivalent method uses the Dirac comb and proceeds by convolving one sinc function with a series of Dirac delta pulses, weighted by the sample values. Neither method is numerically practical. Instead, some type of approximation of the sinc functions, finite in length, is used. The imperfections attributable to the approximation are known as interpolation error.

Practical digital-to-analog converters produce neither scaled and delayed sinc functions, nor ideal Dirac pulses. Instead they produce a piecewise-constant sequence of scaled and delayed rectangular pulses (the zero-order hold), usually followed by a lowpass filter (called an "anti-imaging filter") to remove spurious high-frequency replicas (images) of the original baseband signal.

Aliasing

When $$x(t)$$ is a function with a Fourier transform $$X(f)$$:

 * $$X(f)\ \triangleq\ \int_{-\infty}^{\infty} x(t) \ e^{- i 2 \pi f t} \ {\rm d}t,$$

Then the samples, $$x[n],$$ of $$x(t)$$ are sufficient to create a periodic summation of $$X(f).$$ (see Discrete-time_Fourier_transform):

$$

which is a periodic function and its equivalent representation as a Fourier series, whose coefficients are $$x[n]$$. This function is also known as the discrete-time Fourier transform (DTFT) of the sample sequence.

As depicted, copies of $$X(f)$$ are shifted by multiples of the sampling rate $$f_s = 1/T$$ and combined by addition. For a band-limited function $$(X(f) = 0, \text{ for all } |f| \ge B)$$ and sufficiently large $$f_s,$$ it is possible for the copies to remain distinct from each other. But if the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous $$X(f).$$ Any frequency component above $$f_s/2$$ is indistinguishable from a lower-frequency component, called an alias, associated with one of the copies. In such cases, the customary interpolation techniques produce the alias, rather than the original component. When the sample-rate is pre-determined by other considerations (such as an industry standard), $$x(t)$$ is usually filtered to reduce its high frequencies to acceptable levels before it is sampled. The type of filter required is a lowpass filter, and in this application it is called an anti-aliasing filter.

Derivation as a special case of Poisson summation
When there is no overlap of the copies (also known as "images") of $$X(f)$$, the $$k=0$$ term of $$ can be recovered by the product:

$$X(f) = H(f) \cdot X_{1/T}(f),$$

where:

$$H(f)\ \triangleq\ \begin{cases}1 & |f| < B \\ 0 & |f| > f_s - B. \end{cases}$$

The sampling theorem is proved since $$X(f)$$ uniquely determines $$x(t)$$.

All that remains is to derive the formula for reconstruction. $$H(f)$$ need not be precisely defined in the region $$[B,\ f_s-B]$$ because $$X_{1/T}(f)$$ is zero in that region. However, the worst case is when $$B=f_s/2,$$ the Nyquist frequency. A function that is sufficient for that and all less severe cases is:

$$H(f) = \mathrm{rect} \left(\frac{f}{f_s} \right) = \begin{cases}1 & |f| < \frac{f_s}{2} \\ 0 & |f| > \frac{f_s}{2}, \end{cases}$$

where $$\mathrm{rect}$$ is the rectangular function. Therefore:

 * $$X(f) = \mathrm{rect} \left(\frac{f}{f_s} \right) \cdot X_{1/T}(f)$$
 * $$ = \mathrm{rect}(Tf)\cdot \sum_{n=-\infty}^{\infty} T\cdot x(nT)\ e^{-i 2\pi n T f}$$     (from  $$, above).
 * $$ = \sum_{n=-\infty}^{\infty} x(nT)\cdot \underbrace{T\cdot \mathrm{rect} (Tf) \cdot e^{-i 2\pi n T f}}_{

\mathcal{F}\left \{ \mathrm{sinc} \left( \frac{t - nT}{T} \right) \right \}}.$$

The inverse transform of both sides produces the Whittaker–Shannon interpolation formula:

 * $$x(t) = \sum_{n=-\infty}^{\infty} x(nT)\cdot \mathrm{sinc} \left( \frac{t - nT}{T}\right),$$

which shows how the samples, $$x(nT)$$, can be combined to reconstruct $$x(t)$$.

 * Larger-than-necessary values of $$f_s$$ (smaller values of $$T$$), called oversampling, have no effect on the outcome of the reconstruction and have the benefit of leaving room for a transition band in which $$H(f)$$ is free to take intermediate values. Undersampling, which causes aliasing, is not in general a reversible operation.
 * Theoretically, the interpolation formula can be implemented as a low-pass filter, whose impulse response is $$\mathrm{sinc} (t/T)$$ and whose input is $$\textstyle\sum_{n=-\infty}^{\infty} x(nT)\cdot \delta(t - nT),$$ which is a Dirac comb function modulated by the signal samples. Practical digital-to-analog converters (DAC) implement an approximation like the zero-order hold. In that case, oversampling can reduce the approximation error.

Shannon's original proof
Poisson shows that the Fourier series in $$ produces the periodic summation of $$X(f)$$, regardless of $$f_s$$ and $$B$$. Shannon, however, only derives the series coefficients for the case $$f_s=2B$$. Virtually quoting Shannon's original paper:

 * Let $$X(\omega)$$ be the spectrum of $$x(t).$$ Then

 * $$x(t) = {1 \over 2\pi} \int_{-\infty}^{\infty} X(\omega) e^{i\omega t}\;{\rm d}\omega = {1 \over 2\pi} \int_{-2\pi B}^{2\pi B} X(\omega) e^{i\omega t}\;{\rm d}\omega,$$

 * because $$X(\omega)$$ is assumed to be zero outside the band $$\left|\tfrac{\omega}{2\pi}\right| < B.$$ If we let $$t = \tfrac{n}{2B},$$ where $$n$$ is any positive or negative integer, we obtain:

$$

 * On the left are values of $$x(t)$$ at the sampling points. The integral on the right will be recognized as essentially the $$n^{th}$$ coefficient in a Fourier-series expansion of the function $$X(\omega),$$ taking the interval $$-B$$ to $$B$$ as a fundamental period. This means that the values of the samples $$x(n/2B)$$ determine the Fourier coefficients in the series expansion of $$X(\omega).$$ Thus they determine $$X(\omega),$$ since $$X(\omega)$$ is zero for frequencies greater than $$B,$$ and for lower frequencies $$X(\omega)$$ is determined if its Fourier coefficients are determined. But $$X(\omega)$$ determines the original function $$x(t)$$ completely, since a function is determined if its spectrum is known. Therefore the original samples determine the function $$x(t)$$ completely.

Shannon's proof of the theorem is complete at that point, but he goes on to discuss reconstruction via sinc functions, what we now call the Whittaker–Shannon interpolation formula as discussed above. He does not derive or prove the properties of the sinc function, as the Fourier pair relationship between the rect (the rectangular function) and sinc functions was well known by that time.

"Let $x_n$ be the $n^{th}$ sample. Then the function $x(t)$ is represented by:

 * $x(t) = \sum_{n=-\infty}^{\infty}x_n{\sin(\pi(2Bt-n)) \over \pi(2Bt-n)}.$"

As in the other proof, the existence of the Fourier transform of the original signal is assumed, so the proof does not say whether the sampling theorem extends to bandlimited stationary random processes.

Application to multivariable signals and images

The sampling theorem is usually formulated for functions of a single variable. Consequently, the theorem is directly applicable to time-dependent signals and is normally formulated in that context. However, the sampling theorem can be extended in a straightforward way to functions of arbitrarily many variables. Grayscale images, for example, are often represented as two-dimensional arrays (or matrices) of real numbers representing the relative intensities of pixels (picture elements) located at the intersections of row and column sample locations. As a result, images require two independent variables, or indices, to specify each pixel uniquely—one for the row, and one for the column.

Color images typically consist of a composite of three separate grayscale images, one to represent each of the three primary colors—red, green, and blue, or RGB for short. Other colorspaces using 3-vectors for colors include HSV, CIELAB, XYZ, etc. Some colorspaces such as cyan, magenta, yellow, and black (CMYK) may represent color by four dimensions. All of these are treated as vector-valued functions over a two-dimensional sampled domain.

Similar to one-dimensional discrete-time signals, images can also suffer from aliasing if the sampling resolution, or pixel density, is inadequate. For example, a digital photograph of a striped shirt with high frequencies (in other words, the distance between the stripes is small), can cause aliasing of the shirt when it is sampled by the camera's image sensor. The aliasing appears as a moiré pattern. The "solution" to higher sampling in the spatial domain for this case would be to move closer to the shirt, use a higher resolution sensor, or to optically blur the image before acquiring it with the sensor using an optical low-pass filter.

Another example is shown here in the brick patterns. The top image shows the effects when the sampling theorem's condition is not satisfied. When software rescales an image (the same process that creates the thumbnail shown in the lower image) it, in effect, runs the image through a low-pass filter first and then downsamples the image to result in a smaller image that does not exhibit the moiré pattern. The top image is what happens when the image is downsampled without low-pass filtering: aliasing results.

The sampling theorem applies to camera systems, where the scene and lens constitute an analog spatial signal source, and the image sensor is a spatial sampling device. Each of these components is characterized by a modulation transfer function (MTF), representing the precise resolution (spatial bandwidth) available in that component. Effects of aliasing or blurring can occur when the lens MTF and sensor MTF are mismatched. When the optical image which is sampled by the sensor device contains higher spatial frequencies than the sensor, the under sampling acts as a low-pass filter to reduce or eliminate aliasing. When the area of the sampling spot (the size of the pixel sensor) is not large enough to provide sufficient spatial anti-aliasing, a separate anti-aliasing filter (optical low-pass filter) may be included in a camera system to reduce the MTF of the optical image. Instead of requiring an optical filter, the graphics processing unit of smartphone cameras performs digital signal processing to remove aliasing with a digital filter. Digital filters also apply sharpening to amplify the contrast from the lens at high spatial frequencies, which otherwise falls off rapidly at diffraction limits.

The sampling theorem also applies to post-processing digital images, such as to up or down sampling. Effects of aliasing, blurring, and sharpening may be adjusted with digital filtering implemented in software, which necessarily follows the theoretical principles.

Critical frequency
To illustrate the necessity of $$f_s>2B,$$ consider the family of sinusoids generated by different values of $$\theta$$ in this formula:

 * $$x(t) = \frac{\cos(2 \pi B t + \theta )}{\cos(\theta )}\ = \ \cos(2 \pi B t) - \sin(2 \pi B t)\tan(\theta ), \quad -\pi/2 < \theta < \pi/2.$$

With $$f_s=2B$$ or equivalently $$T=1/2B,$$ the samples are given by:

 * $$x(nT) = \cos(\pi n) - \underbrace{\sin(\pi n)}_{0}\tan(\theta ) = (-1)^n$$

That sort of ambiguity is the reason for the strict inequality of the sampling theorem's condition.

Sampling of non-baseband signals
As discussed by Shannon:

"A similar result is true if the band does not start at zero frequency but at some higher value, and can be proved by a linear translation (corresponding physically to single-sideband modulation) of the zero-frequency case. In this case the elementary pulse is obtained from $\sin(x)/x$ by single-side-band modulation."

That is, a sufficient no-loss condition for sampling signals that do not have baseband components exists that involves the width of the non-zero frequency interval as opposed to its highest frequency component. See sampling for more details and examples.

For example, in order to sample FM radio signals in the frequency range of 100–102 MHz, it is not necessary to sample at 204 MHz (twice the upper frequency), but rather it is sufficient to sample at 4 MHz (twice the width of the frequency interval).

A bandpass condition is that $$X(f) = 0,$$ for all nonnegative $$f$$ outside the open band of frequencies:
 * $$\left(\frac{N}2 f_\mathrm{s}, \frac{N+1}2 f_\mathrm{s}\right),$$

for some nonnegative integer $$N$$. This formulation includes the normal baseband condition as the case $$N=0.$$

The corresponding interpolation function is the impulse response of an ideal brick-wall bandpass filter (as opposed to the ideal brick-wall lowpass filter used above) with cutoffs at the upper and lower edges of the specified band, which is the difference between a pair of lowpass impulse responses:

$$(N+1)\,\operatorname{sinc} \left(\frac{(N+1)t}T\right) - N\,\operatorname{sinc}\left( \frac{Nt}T \right).$$

Other generalizations, for example to signals occupying multiple non-contiguous bands, are possible as well. Even the most generalized form of the sampling theorem does not have a provably true converse. That is, one cannot conclude that information is necessarily lost just because the conditions of the sampling theorem are not satisfied; from an engineering perspective, however, it is generally safe to assume that if the sampling theorem is not satisfied then information will most likely be lost.

Nonuniform sampling
The sampling theory of Shannon can be generalized for the case of nonuniform sampling, that is, samples not taken equally spaced in time. The Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition. Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction.

The general theory for non-baseband and nonuniform samples was developed in 1967 by Henry Landau. He proved that the average sampling rate (uniform or otherwise) must be twice the occupied bandwidth of the signal, assuming it is a priori known what portion of the spectrum was occupied.

In the late 1990s, this work was partially extended to cover signals for which the amount of occupied bandwidth is known but the actual occupied portion of the spectrum is unknown. In the 2000s, a complete theory was developed (see the section Sampling below the Nyquist rate under additional restrictions below) using compressed sensing. In particular, the theory, using signal processing language, is described in a 2009 paper by Mishali and Eldar. They show, among other things, that if the frequency locations are unknown, then it is necessary to sample at least at twice the Nyquist criteria; in other words, you must pay at least a factor of 2 for not knowing the location of the spectrum. Note that minimum sampling requirements do not necessarily guarantee stability.

Sampling below the Nyquist rate under additional restrictions
The Nyquist–Shannon sampling theorem provides a sufficient condition for the sampling and reconstruction of a band-limited signal. When reconstruction is done via the Whittaker–Shannon interpolation formula, the Nyquist criterion is also a necessary condition to avoid aliasing, in the sense that if samples are taken at a slower rate than twice the band limit, then there are some signals that will not be correctly reconstructed. However, if further restrictions are imposed on the signal, then the Nyquist criterion may no longer be a necessary condition.

A non-trivial example of exploiting extra assumptions about the signal is given by the recent field of compressed sensing, which allows for full reconstruction with a sub-Nyquist sampling rate. Specifically, this applies to signals that are sparse (or compressible) in some domain. As an example, compressed sensing deals with signals that may have a low overall bandwidth (say, the effective bandwidth $$EB$$) but the frequency locations are unknown, rather than all together in a single band, so that the passband technique does not apply. In other words, the frequency spectrum is sparse. Traditionally, the necessary sampling rate is thus $$2B.$$ Using compressed sensing techniques, the signal could be perfectly reconstructed if it is sampled at a rate slightly lower than $$2EB.$$ With this approach, reconstruction is no longer given by a formula, but instead by the solution to a linear optimization program.

Another example where sub-Nyquist sampling is optimal arises under the additional constraint that the samples are quantized in an optimal manner, as in a combined system of sampling and optimal lossy compression. This setting is relevant in cases where the joint effect of sampling and quantization is to be considered, and can provide a lower bound for the minimal reconstruction error that can be attained in sampling and quantizing a random signal. For stationary Gaussian random signals, this lower bound is usually attained at a sub-Nyquist sampling rate, indicating that sub-Nyquist sampling is optimal for this signal model under optimal quantization.

Historical background
The sampling theorem was implied by the work of Harry Nyquist in 1928, in which he showed that up to $$2B$$ independent pulse samples could be sent through a system of bandwidth $$B$$; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step-response sine integral; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English).

The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon. The mathematician E. T. Whittaker published similar results in 1915, J. M. Whittaker in 1935, and Gabor in 1946 ("Theory of communication").

In 1948 and 1949, Claude E. Shannon published the two revolutionary articles in which he founded the information theory. In Shannon 1948 the sampling theorem is formulated as "Theorem 13": Let $$f(t)$$ contain no frequencies over W. Then

$$f(t) = \sum_{n=-\infty}^\infty X_n \frac{\sin \pi(2Wt - n)}{\pi(2Wt - n)},$$ where $$X_n = f\left(\frac n {2W} \right).$$

It was not until these articles were published that the theorem known as "Shannon's sampling theorem" became common property among communication engineers, although Shannon himself writes that this is a fact which is common knowledge in the communication art. A few lines further on, however, he adds: "but in spite of its evident importance, [it] seems not to have appeared explicitly in the literature of communication theory". Despite his sampling theorem being published at the end of the 1940s, Shannon had derived his sampling theorem as early as 1940.

Other discoverers
Others who have independently discovered or played roles in the development of the sampling theorem have been discussed in several historical articles, for example, by Jerri and by Lüke. For example, Lüke points out that H. Raabe, an assistant to Küpfmüller, proved the theorem in his 1939 Ph.D. dissertation; the term Raabe condition came to be associated with the criterion for unambiguous representation (sampling rate greater than twice the bandwidth). Meijering mentions several other discoverers and names in a paragraph and pair of footnotes:

"As pointed out by Higgins, the sampling theorem should really be considered in two parts, as done above: the first stating the fact that a bandlimited function is completely determined by its samples, the second describing how to reconstruct the function using its samples. Both parts of the sampling theorem were given in a somewhat different form by J. M. Whittaker and before him also by Ogura. They were probably not aware of the fact that the first part of the theorem had been stated as early as 1897 by Borel. As we have seen, Borel also used around that time what became known as the cardinal series. However, he appears not to have made the link. In later years it became known that the sampling theorem had been presented before Shannon to the Russian communication community by Kotel'nikov. In more implicit, verbal form, it had also been described in the German literature by Raabe. Several authors have mentioned that Someya introduced the theorem in the Japanese literature parallel to Shannon. In the English literature, Weston introduced it independently of Shannon around the same time."

- Eric Meijering, "A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing" (citations omitted)

In Russian literature it is known as the Kotelnikov's theorem, named after Vladimir Kotelnikov, who discovered it in 1933.

Why Nyquist?
Exactly how, when, or why Harry Nyquist had his name attached to the sampling theorem remains obscure. The term Nyquist Sampling Theorem (capitalized thus) appeared as early as 1959 in a book from his former employer, Bell Labs, and appeared again in 1963, and not capitalized in 1965. It had been called the Shannon Sampling Theorem as early as 1954, but also just the sampling theorem by several other books in the early 1950s.

In 1958, Blackman and Tukey cited Nyquist's 1928 article as a reference for the sampling theorem of information theory, even though that article does not treat sampling and reconstruction of continuous signals as others did. Their glossary of terms includes these entries:

"Sampling theorem (of information theory): Nyquist's result that equi-spaced data, with two or more points per cycle of highest frequency, allows reconstruction of band-limited functions. (See Cardinal theorem.) Cardinal theorem (of interpolation theory): A precise statement of the conditions under which values given at a doubly infinite set of equally spaced points can be interpolated to yield a continuous band-limited function with the aid of the function $\frac{\sin (x - x_i)}{x - x_i}.$"

Exactly what "Nyquist's result" they are referring to remains mysterious.

When Shannon stated and proved the sampling theorem in his 1949 article, according to Meijering, "he referred to the critical sampling interval $$T = \frac 1 {2W}$$ as the Nyquist interval corresponding to the band $$W,$$ in recognition of Nyquist's discovery of the fundamental importance of this interval in connection with telegraphy". This explains Nyquist's name on the critical interval, but not on the theorem.

Similarly, Nyquist's name was attached to Nyquist rate in 1953 by Harold S. Black:

"If the essential frequency range is limited to $B$ cycles per second, $2B$ was given by Nyquist as the maximum number of code elements per second that could be unambiguously resolved, assuming the peak interference is less than half a quantum step. This rate is generally referred to as signaling at the Nyquist rate and $\frac 1 {2B}$ has been termed a Nyquist interval."

According to the Oxford English Dictionary, this may be the origin of the term Nyquist rate. In Black's usage, it is not a sampling rate, but a signaling rate.

title: Nyquist–Shannon sampling theorem


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Nicene and Post-Nicene Fathers

A Select Library of the Nicene and Post-Nicene Fathers of the Christian Church, usually known as the Nicene and Post-Nicene Fathers (NPNF), is a set of books containing translations of early Christian writings into English. It was published between 1886 and 1900.

Unlike the Ante-Nicene Fathers which was produced by using earlier translations of the Ante-Nicene Christian Library (ANCL), the Nicene and Post-Nicene Fathers was printed simultaneously in Europe and in America, by T. & T. Clark, by Christian Literature Company and other American editors. The translations were in the main revised versions of those made for the Oxford Movement Library of the Fathers series. T. & T. Clark was surely convinced by the commercial success of the cheaper American version/revision of the ANCL, although of lesser quality on some minor points. The Swiss-born, German-educated Philip Schaff was commissioned to supervise the first series of the NPNF. He was joined by the British Henry Wace for the second series.

The volumes include the following:

Augustine volumes

 * Volume I. Prolegomena: St. Augustine's Life and Work, Confessions, Letters
 * Volume II. The City of God, Christian Doctrine
 * Volume III. On the Holy Trinity, Doctrinal Treatises, Moral Treatises
 * On the Trinity. The Enchiridion. On the Catechising of the Uninstructed. A Treatise on Faith and the Creed. Concerning Faith of Things Not Seen. On the Profit of Believing. On the Creed: a Sermon to the Catechumens. On Continence. On the Good of Marriage. Of Holy Virginity. On the Good of Widowhood. On Lying. Against Lying. Of the Work of Monks. On Patience. On Care to Be Had for the Dead.
 * Volume IV.  The Anti-Manichaean Writings, The Anti-Donatist Writings
 * Of the Morals of the Catholic Church. On the Morals of the Manichaeans. Concerning Two Souls, Against the Manichaeans. Acts or Disputation Against Fortunatus, the Manichaean. Against the Epistle of Manichaeus Called Fundamental. Reply to Faustus the Manichaean (33 books). Concerning the Nature of Good, Against the Manichaeans. On Baptism, Against the Donatists (7 books). In Answer to the Letters of Petilian, the Donatist, Bishop of Cirta (3 books). A Treatise Concerning the Correction of the Donatists.
 * Volume V.  Anti-Pelagian Writings
 * On the Merits and Forgiveness of Sins, and on the Baptism of Infants (3 books). On the Spirit and the Letter. On Nature and Grace, Against Pelagius. Concerning Man's Perfection in Righteousness. On the Proceedings of Pelagius. A Treatise on the Grace of Christ, and on Original Sin. On Marriage and Concupiscence (2 books). On the Soul and Its Origin (4 books). A Treatise Against Two Letters of the Pelagians (4 books). A Treatise on Grace and Free Will. Treatise on Rebuke and Grace. A Treatise on the Predestination of the Saints. A Treatise on the Gift of Perseverance, Being the Second Book of the Predestination of the Saints.
 * Volume VI.  Sermon on the Mount, Harmony of the Gospels, Homilies on the Gospels
 * Our Lord's Sermon on the Mount. The Harmony of the Gospels. 97 Sermons on Selected Lessons of the New Testament.
 * Volume VII.  Homilies on the Gospel of John, Homilies on the First Epistle of John, Soliloquies
 * Tractates on John (124 tractates). Ten Homilies on the First Epistle of John. Two Books of Soliloquies.
 * Volume VIII.  Expositions on the Psalms

Chrysostom volumes

 * Volume IX.  On the Priesthood, ascetic Treatises, Select Homilies and Letters, Homilies on the Statues
 * Treatise on the Priesthood. An Exhortation to Theodore After His Fall. Letter I. Letter II. Letter to a Young Widow. Homilies on S. Ignatius and S. Babylas. Eulogy. On the Holy Martyr, S. Babylas. Concerning Lowliness of Mind. Instructions to Catechumens. Three Homilies Concerning the Power of Demons. Against Marcionists and Manichaeans. Homily on the Paralytic Let Down Through the Roof. To Those Who Had Not Attended the Assembly. Homily Against Publishing the Errors of the Brethren. On Eutropius, Patrician and Consul. Two Homilies on Eutropius. A Treatise to Prove that No One Can Harm the Man Who Does Not Injure Himself. Letters to Olympias. To Castus, Valerius, Diophantus, Cyriacus. Correspondence of St. Chrysostom, and the Church at Constantinople, with Innocent, Bishop of Rome. Homilies Concerning the Statues.
 * Volume X.  Homilies on the Gospel of St. Matthew
 * Volume XI.  Homilies on the Acts of the Apostles and the Epistle to the Romans
 * Volume XII.  Homilies on First and Second Corinthians
 * Volume XIII.  Homilies on the Epistles to the Galatians, Ephesians, Philippians, Colossians, Thessalonians, Timothy, Titus, and Philemon
 * Volume XIV.  Homilies on the Gospel of St. John and the Epistle to the Hebrews

title: Nicene and Post-Nicene Fathers


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Porphyry

Porphyry (Πορφύριος, Porphyrios "purple-clad") may refer to:

Geology

 * Porphyry (geology), an igneous rock with large crystals in a fine-grained matrix, often purple, and prestigious Roman sculpture material
 * Shoksha porphyry, quartzite of purple color resembling true porphyry mined near the village of Shoksha, Karelia, Russia
 * Porphyritic, the general igneous texture of a rock with two distinct crystal (phenocryst) sizes
 * Porphyry copper deposit, a primary (low grade) ore deposit of copper, consisting of porphyry rocks

Places

 * Mons Porphyrites, the only porphyry quarry worked in the ancient world for the emperor's building works and statuary
 * Porphyry Island, in Lake Superior, Canada
 * Porphyry Mountain in Alaska, United States
 * Porphyry, a system of astrological house division
 * Porphyry, a vineyard near Seaham, New South Wales, Australia

Animals and plants

 * Porphyra, a foliose red algal genus of laver
 * Oliva porphyria, a species of sea snail
 * Porphyrio, the swamphens, a genus of birds in the rail family
 * Porphyrios (whale), a 6th-century whale

People and characters

 * Porphyrion, a giant in Greek mythology
 * Pomponius Porphyrion (fl. 2nd or 3rd century), Latin grammarian
 * Porphyry (philosopher) (234–305), Neoplatonic philosopher
 * Porphyrios of Philippi (fl. early 4th century), bishop of the Metropolis of Philippi, Neapolis and Thasos
 * Publilius Optatianus Porfyrius, Latin poet
 * Porphyry of Gaza, or Saint Porphyrius (347–420), Bishop of Gaza
 * Porphyrius the Charioteer (fl. 5th/6th centuries), Roman charioteer
 * Porphyrios Dikaios (1904–1971), Greek Cypriot archaeologist
 * Porphyrios of Kafsokalyvia (1906–1991), Greek Athonite hieromonk and Eastern Orthodox saint
 * Demetri Porphyrios (born 1949), Greek architect
 * Porfirije, Serbian Patriarch (born 1961), 46th Patriarch of the Serbian Orthodox Church

title: Porphyry


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Pope Leo XI

Pope Leo XI (Leone XI; 2 June 1535 – 27 April 1605), born Alessandro Ottaviano de' Medici, was head of the Catholic Church and ruler of the Papal States from 1 April 1605 to his death, on 27 April 1605. His pontificate is one of the briefest in history, having lasted under a month. He was from the prominent House of Medici originating from Florence. Medici's mother opposed his entering the priesthood and sought to prevent it by having him given secular honours, but after her death he eventually was ordained a priest in 1567. In his career he served as Florence's ambassador to the pope, Bishop of Pistoia, Archbishop of Florence, Papal legate to France, and as the cardinal Prefect for the Congregation of Bishops and Regulars. He was elected to the papacy in the March 1605 papal conclave and served as pope for 27 days.

Early life
Alessandro Ottaviano de' Medici was born in Florence as the son of Ottaviano de' Medici and Francesca Salviati. His family belonged to Medici di Ottajano, a cadet branch of the House of Medici. He was also the great-nephew of Pope Leo X. Ottaviano died early in his son’s life, and thereafter Alessandro was home schooled by a Dominican priest, Vincenzo Ercolano.

Alessandro felt the call to the priesthood, but his mother opposed this since he was the only male in the family. She sent him instead to the court of the Grand Duke of Tuscany, who appointed him a knight of San Stefano. In 1560 he travelled to Rome where he commenced a lifelong friendship and collaboration with Philip Neri, the future saint. It was Philip who predicted that Alessandro would ascend to the pontificate. His mother died in 1566, at which point he resumed his studies to become a priest. His ordination took place on 22 July 1567.

Priesthood

Alessandro served as the Florentine ambassador to Pope Pius V from 1569 to 1584 and in 1573 was appointed by Pope Gregory XIII Bishop of Pistoia. In March 1573 he received episcopal consecration in Rome. In 1574 he was made Archbishop of Florence.

In 1583 he was made a cardinal by Pope Sixtus V and on 9 January 1584 received the title of Cardinal-Priest of Santi Quirico e Giulitta, after a titular church previously known as San Ciriaco alle Terme Diocleziane. In later years, according to custom he would opt for other titular churches.

In 1596 Pope Clement VIII sent Alessandro as a papal legate to France, where he remained until 1598, when he received word of his appointment as Prefect of the Congregation of Bishops and Regulars.

Papal election
On 14 March 1605, eleven days after the death of Clement VIII, 62 cardinals entered the conclave. Prominent among the candidates for the papacy were the great historian Caesar Baronius and the famous Jesuit controversialist Robert Bellarmine, future saint.

But Pietro Aldobrandini, the leader of the Italian party among the cardinals, allied with the French cardinals and brought about the election of Alessandro against the express wish of King Philip III of Spain. King Henry IV of France is said to have spent 300,000 écus in the promotion of Alessandro's candidacy.

On 1 April 1605, Cardinal Alessandro de' Medici was elected as pope. He chose to be called Leo XI in honor of his uncle Pope Leo X. He was crowned on 10 April 1605 by the protodeacon, Cardinal Francesco Sforza and he took possession of the Basilica of Saint John Lateran on 17 April 1605.

Death
When he was elected, Leo XI was almost 70 years of age, and he died 27 days later. His death came as a result of fatigue and cold in the ceremony of taking possession of the Basilica of St John Lateran on 17 April; he started suffering from a fever the following day.

title: Pope Leo XI


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Pope Leo X

Pope Leo X (Leone X; born Giovanni di Lorenzo de' Medici, 11 December 1475 –1 December 1521) was head of the Catholic Church and ruler of the Papal States from 9 March 1513 to his death, in December 1521.

Born into the prominent political and banking Medici family of Florence, Giovanni was the second son of Lorenzo de' Medici, ruler of the Florentine Republic, and was elevated to the cardinalate in 1489. Following the death of Pope Julius II, Giovanni was elected pope after securing the backing of the younger members of the College of Cardinals. Early on in his rule he oversaw the closing sessions of the Fifth Council of the Lateran, but struggled to implement the reforms agreed. In 1517 he led a costly war that succeeded in securing his nephew Lorenzo di Piero de' Medici as Duke of Urbino, but reduced papal finances.

In Protestant circles, Leo is associated with granting indulgences for those who donated to reconstruct St. Peter's Basilica, a practice that was soon challenged by Martin Luther's 95 Theses. Leo rejected the Protestant Reformation, and his Papal bull of 1520, Exsurge Domine, condemned Luther's condemnatory stance, rendering ongoing communication difficult.

He borrowed and spent money without circumspection and was a significant patron of the arts. Under his reign, Marco Girolamo Vida began composing at the Pope's request a Virgilian Latin epic poem about the life of Jesus called the Christiad, progress was made on the rebuilding of St. Peter's Basilica, and artists such as Raphael decorated the Vatican rooms. Leo also reorganised the Roman University, and promoted Renaissance humanist study of literature, poetry, and Classics. He died in 1521 and is buried in Santa Maria sopra Minerva, Rome. He was the last pope not to have been in priestly orders at the time of his election to the papacy.

Early life

Giovanni di Lorenzo de' Medici was born on 11 December 1475 in Florence, the second son of Lorenzo the Magnificent, head of the Florentine Republic, and Clarice Orsini. From an early age Giovanni was destined for an ecclesiastical career. He received the tonsure at the age of seven and was soon granted rich benefices and preferments.

His father, Lorenzo de' Medici, was worried about his character early on and wrote a letter to Giovanni to warn him to avoid vice and luxury at the beginning of his ecclesiastical career. Here is a notable excerpt: "There is one rule which I would recommend to your attention in preference to all others. Rise early in the morning. This will not only contribute to your health, but will enable you to arrange and expedite the business of the day; and as there are various duties incident to".

Cardinal
His father prevailed on his relative Pope Innocent VIII to name him cardinal of Santa Maria in Domnica on 9 March 1489 when he was age 13, although he was not allowed to wear the insignia or share in the deliberations of the college until three years later. Meanwhile, he received an education at Lorenzo's humanistic court under such men as Angelo Poliziano, Pico della Mirandola, Marsilio Ficino and Bernardo Dovizio Bibbiena. From 1489 to 1491 he studied theology and canon law at Pisa.

On 23 March 1492, he was formally admitted into the Sacred College of Cardinals and took up his residence at Rome, receiving a letter of advice from his father. The death of Lorenzo on the following 8 April temporarily recalled the 16-year-old Giovanni to Florence. He returned to Rome to participate in the conclave of 1492 which followed the death of Innocent VIII, and unsuccessfully opposed the election of Cardinal Rodrigo Borgia as Pope Alexander VI.

He subsequently made his home with his elder brother Piero in Florence throughout the agitation of Girolamo Savonarola and the invasion of Charles VIII of France, until the uprising of the Florentines and the expulsion of the Medici in November 1494. While Piero found refuge at Venice and Urbino, Giovanni traveled in Germany, in the Netherlands, and in France.

In May 1500, he returned to Rome, where he was received with outward cordiality by Pope Alexander VI, and where he lived for several years immersed in art and literature. In 1503 he welcomed the accession of Pope Julius II to the pontificate; the death of Piero de' Medici in the same year made Giovanni head of his family. On 1 October 1511, he was appointed papal legate of Bologna and the Romagna, and when the Florentine republic declared in favour of the schismatic Pisans, Julius II sent Giovanni (as legate) with the Papal army venturing against the French. The French won a major battle and captured Giovanni. This and other attempts to regain political control of Florence were frustrated until a bloodless revolution permitted the return of the Medici. Giovanni's younger brother Giuliano was placed at the head of the republic, but Giovanni managed the government.

Papal election
Giovanni was elected pope on 9 March 1513, and this was proclaimed two days later. The absence of the French cardinals effectively reduced the election to a contest between Giovanni (who had the support of the younger and noble members of the college) and Raffaele Riario (who had the support of the older group). On 15 March 1513, he was ordained priest, and consecrated as bishop on 17 March. He was crowned pope on 19 March 1513 at the age of 37. He was the last non-priest to be elected pope.

War of Urbino
Leo had intended his younger brother Giuliano and his nephew Lorenzo for brilliant secular careers. He had named them Roman patricians; the latter he had placed in charge of Florence; the former, for whom he planned to carve out a kingdom in central Italy of Parma, Piacenza, Ferrara and Urbino, he had taken with himself to Rome and married to Filiberta of Savoy.

The death of Giuliano in March 1516, however, caused the pope to transfer his ambitions to Lorenzo. At the very time (December 1516) that peace between France, Spain, Venice and the Habsburg monarchy seemed to give some promise of a Christendom united against the Turks, Leo obtained 150,000 ducats towards the expenses of the expedition from Henry VIII of England, in return for which he entered the imperial league of Spain and England against France.

The war lasted from February to September 1517 and ended with the expulsion of the duke and the triumph of Lorenzo; but it revived the policy of Alexander VI, increased brigandage and anarchy in the Papal States, hindered the preparations for a crusade and wrecked the papal finances. Francesco Guicciardini reckoned the cost of the war to Leo at the sum of 800,000 ducats. Ultimately, however, Lorenzo was confirmed as the new duke of Urbino.

Plans for a crusade
The War of Urbino was further marked by a crisis in the relations between the pope and the cardinals. The sacred college had allegedly grown very worldly and troublesome since the time of Sixtus IV, and Leo took advantage of a plot by several of its members to poison him, not only to inflict exemplary punishments by executing one (Alfonso Petrucci) and imprisoning several others, but also to make radical changes in the college.

On 3 July 1517, he published the names of thirty-one new cardinals, a number almost unprecedented in the history of the papacy. Among the nominations were such notable men such as Lorenzo Campeggio, Giovanni Battista Pallavicino, Adrian of Utrecht, Thomas Cajetan, Cristoforo Numai and Egidio Canisio. The naming of seven members of prominent Roman families, however, reversed the policy of his predecessor which had kept the political factions of the city out of the Curia. Other promotions were for political or family considerations or to secure money for the war against Urbino. The pope was accused of having exaggerated the conspiracy of the cardinals for purposes of financial gain, but most of such accusations appear unsubstantiated.

Leo, meanwhile, felt the need of staying the advance of the Ottoman sultan, Selim I, who was threatening eastern Europe, and made elaborate plans for a crusade. A truce was to be proclaimed throughout Christendom; the pope was to be the arbiter of disputes; the emperor and the king of France were to lead the army; England, Spain and Portugal were to furnish the fleet; and the combined forces were to be directed against Constantinople. Papal diplomacy in the interests of peace failed, however; Cardinal Thomas Wolsey made England, not the pope, the arbiter between France and the Empire; and much of the money collected for the crusade from tithes and indulgences was spent in other ways.

In 1519 Hungary concluded a three years' truce with Selim I, but the succeeding sultan, Suleiman the Magnificent, renewed the war in June 1521 and on 28 August captured the citadel of Belgrade. The pope was greatly alarmed, and although he was then involved in war with France he sent about 30,000 ducats to the Hungarians. Leo treated the Eastern Catholic Greeks with great loyalty, and by a bull of 18 May 1521 forbade Latin clergy to celebrate mass in Greek churches and Latin bishops to ordain Greek clergy. These provisions were later strengthened by Clement VII and Paul III and went far to settle the constant disputes between the Latins and Uniate Greeks.

Protestant Reformation
Leo was disturbed throughout his pontificate by schism, especially the Reformation sparked by Martin Luther.

In response to concerns about misconduct from some indulgence preachers, in 1517 Martin Luther wrote his Ninety-five Theses on the topic of indulgences. The resulting pamphlet spread Luther's ideas throughout Germany and Europe. Leo failed to fully comprehend the importance of the movement, and in February 1518 he directed the vicar-general of the Augustinians to impose silence on his monks.

On 24 May, Luther sent an explanation of his theses to the pope; on 7 August he was summoned to appear at Rome. An arrangement was effected, however, whereby that summons was cancelled, and Luther went instead to Augsburg in October 1518 to meet the papal legate, Cardinal Cajetan; but neither the arguments of the cardinal, nor Leo's dogmatic papal bull of 9 November requiring all Christians to believe in the pope's power to grant indulgences, moved Luther to retract. A year of fruitless negotiations followed, during which the controversy took popular root across the German states.

A further papal bull of 15 June 1520, Exsurge Domine or Arise, O Lord, condemned forty-one propositions extracted from Luther's teachings, and was taken to Germany by Johann Eck in his capacity as apostolic nuncio. Leo followed by formally excommunicating Luther by the bull Decet Romanum Pontificem or It Befits the Roman Pontiff, on 3 January 1521. In a brief, the Pope also directed Charles V, Holy Roman Emperor to take energetic measures against heresy.

Leo was pope during the spread of Lutheranism into Scandinavia. The pope had repeatedly used the rich northern benefices to reward members of the Roman curia, and towards the close of the year 1516, he sent the impolitic Giovanni Angelo Arcimboldi as papal nuncio to Denmark to collect money for St Peter's. This led to the Reformation in Denmark. King Christian II took advantage of the growing dissatisfaction of the native clergy toward the papal government, and of Arcimboldi's interference in the Swedish revolt, to expel the nuncio and summon Lutheran theologians to Copenhagen in 1520. Christian approved a plan by which a formal state church should be established in Denmark, all appeals to Rome should be abolished, and the king and diet should have final jurisdiction in ecclesiastical causes. Leo sent a new nuncio to Copenhagen (1521) in the person of the Minorite Francesco de Potentia, who readily absolved the king and received the bishopric of Skara. The pope or his legate, however, took no steps to correct abuses or otherwise discipline the Scandinavian churches.

Consistories
The pope created 42 new cardinals in eight consistories including two cousins (one who would become his successor Pope Clement VII) and a nephew. He also elevated Adriaan Florensz Boeyens into the cardinalate who would become his immediate successor Pope Adrian VI. Leo X's consistory of 1 July 1517 saw 31 cardinals created, and this remained the largest allocation of cardinals in one consistory until Pope John Paul II named 42 cardinals in 2001.

Canonizations
Pope Leo X canonized eleven individuals during his reign with seven of those being a group cause of martyrs. The most notable canonization from his papacy was that of Francis of Paola on 1 May 1519.

Final years
That Leo did not do more to check the anti-papal rebellion in Germany and Scandinavia is to be partially explained by the political complications of the time, and by his own preoccupation with papal and Medicean politics in Italy. The death of the Emperor Maximilian in 1519 had seriously affected the situation. Leo vacillated between the powerful candidates for the succession, allowing it to appear at first that he favoured Francis or a minor German prince. He finally accepted Charles of Spain as inevitable.

Leo was now eager to unite Ferrara, Parma and Piacenza to the Papal States. An attempt late in 1519 to seize Ferrara failed, and the pope recognized the need for foreign aid. In May 1521 a treaty of alliance was signed at Rome between him and the emperor. Milan and Genoa were to be taken from France and restored to the Empire, and Parma and Piacenza were to be given to the Church on the expulsion of the French. The expense of enlisting 10,000 Swiss was to be borne equally by Pope and emperor. Charles V took Florence and the Medici family under his protection and promised to punish all enemies of the Catholic faith. Leo agreed to invest Charles V with the Kingdom of Naples, to crown him Holy Roman Emperor, and to aid in a war against Venice. It was provided that England and the Swiss might also join the league. Henry VIII announced his adherence in August 1521. Francis I of France had already begun war with Charles V in Navarre, and in Italy, too, the French made the first hostile movement on 23 June 1521. Leo at once announced that he would excommunicate the king of France and release his subjects from their allegiance unless Francis I laid down his arms and surrendered Parma and Piacenza to the Church. The pope lived to hear the joyful news of the capture of Milan from the French and of the occupation by papal troops of the long-coveted provinces (November 1521).

Having fallen ill with bronchopneumonia, Pope Leo X died on 1 December 1521, so suddenly that the last sacraments could not be administered; but the contemporary suspicions of poison were unfounded. He was buried in Santa Maria sopra Minerva.

General assessment
Leo has been criticized for his handling of the events of the papacy. He had a musical and pleasant voice and a cheerful temper. He was eloquent in speech and elegant in his manners and epistolary style. He enjoyed music and the theatre, art and poetry, the masterpieces of the ancients and the creations of his contemporaries, especially those seasoned with wit and learning. He especially delighted in ex tempore Latin verse-making (at which he excelled) and cultivated improvisatori. He is said to have stated, "Let us enjoy the papacy since God has given it to us", Ludwig von Pastor says that it is by no means certain that he made the remark; and historian Klemens Löffler says that "the Venetian ambassador who related this of him was not unbiased, nor was he in Rome at the time." However, there is no doubt that he was by nature pleasure-loving and that the anecdote reflects his casual attitude to the high and solemn office to which he had been called. On the other hand, in spite of his worldliness, Leo prayed, fasted, went to confession before celebrating Mass in public, and conscientiously participated in the religious services of the Church. To the virtues of liberality, charity, and clemency he added the Machiavellian qualities of deception and shrewdness, so highly esteemed by the princes of his time.

The character of Leo X was formerly assailed by lurid aspersions of debauchery, murder, impiety, and atheism. In the 17th century it was estimated that 300 or 400 writers, more or less, reported (on the authority of a single polemical anti-Catholic source) a story that when someone had quoted to Leo a passage from one of the Four Evangelists, he had replied that it was common knowledge "how profitable that fable of Christe hath ben to us and our companie". These aspersions and more were examined by William Roscoe in the 19th century (and again by Ludwig von Pastor in the 20th) and rejected. Nevertheless, even the eminent philosopher David Hume, while claiming that Leo was too intelligent to believe in Catholic doctrine, conceded that he was "one of the most illustrious princes that ever sat on the papal throne. Humane, beneficent, generous, affable; the patron of every art, and friend of every virtue". Martin Luther, in a conciliatory letter to Leo, himself testified to Leo's universal reputation for morality: "Indeed, the published opinion of so many great men and the repute of your blameless life are too widely famed and too much reverenced throughout the world to be assailed by any man, of however great name, or by any arts. I am not so foolish to attack one whom everybody praises ..."

The final report of the Venetian ambassador Marino Giorgi supports Hume's assessment of affability and testifies to the range of Leo's talents. Bearing the date of March 1517 it indicates some of his predominant characteristics: "The pope is a good-natured and extremely free-hearted man, who avoids every difficult situation and above all wants peace; he would not undertake a war himself unless his own personal interests were involved; he loves learning; of canon law and literature he possesses remarkable knowledge; he is, moreover, a very excellent musician."

Leo is the fifth of the six popes who are unfavourably profiled by historian Barbara Tuchman in The March of Folly, and who are accused by her of precipitating the Protestant Reformation. Tuchman describes Leo as a cultured – if religiously devout – hedonist.

Intellectual interests
Leo X's love for all forms of art stemmed from the humanistic education he received in Florence, his studies in Pisa and his extensive travel throughout Europe when a youth. He loved the Latin poems of the humanists, the tragedies of the Greeks and the comedies of Cardinal Bibbiena and Ariosto, while relishing the accounts sent back by the explorers of the New World. Yet "Such a humanistic interest was itself religious. ... In the Renaissance, the vines of the classical world and the Christian world, of Rome, were seen as intertwined. It was a historically minded culture where artists' representations of Cupid and the Madonna, of Hercules and St. Peter could exist side-by-side".

Love of music
Pastor says that "From his youth, Leo, who had a fine ear and a melodious voice, loved music to the pitch of fanaticism". As pope he procured the services of professional singers, instrumentalists and composers from as far away as France, Germany and Spain. Next to goldsmiths, the highest salaries recorded in the papal accounts are those paid to musicians, who also received largesse from Leo's private purse. Their services were retained not so much for the delectation of Leo and his guests at private social functions as for the enhancement of religious services on which the pope placed great store. The standard of singing of the papal choir was a particular object of Leo's concern, with French, Dutch, Spanish and Italian singers being retained. Large sums of money were also spent on the acquisition of highly ornamented musical instruments, and he was especially assiduous in securing musical scores from Florence. He also fostered technical improvements developed for the diffusion of such scores. Ottaviano Petrucci, who had overcome practical difficulties in the way of using movable type to print musical notation, obtained from Leo X the exclusive privilege of printing organ scores (which, according to the papal brief, "adds greatly to the dignity of divine worship") for a period for 15 years from 22 October 1513. In addition to fostering the performance of sung Masses, he promoted the singing of the Gospel in Greek in his private chapel.

Unworthy pursuits
Even those who defend him against the more outlandish attacks on his character acknowledge that he partook of entertainment such as masquerades, "jests," fowling, and hunting boar and other wild beasts. According to one biographer, he was "engrossed in idle and selfish amusements".

Leo indulged buffoons at his Court, but also tolerated behaviour which made them the object of ridicule. One case concerned the conceited improvisatore Giacomo Baraballo, Abbot of Gaeta, who was the butt of a burlesque procession organised in the style of an ancient Roman triumph. Baraballo was dressed in festal robes of velvet and with ermine and presented to the pope. He was then taken to the piazza of St Peter's and was mounted on the back of Hanno, a white elephant, the gift of King Manuel I of Portugal. The magnificently ornamented animal was then led off in the direction of the Capitol to the sound of drums and trumpets. But while crossing the bridge of Sant'Angelo over the Tiber, the elephant, already distressed by the noise and confusion around him, shied violently, throwing his passenger onto the muddy riverbank below.

Sexuality
Leo's most recent biographer, Carlo Falconi, says Leo hid a private life of moral irregularity behind a mask of urbanity. Scabrous verse libels of the type known as pasquinades were particularly abundant during the conclave which followed Leo's death in 1521 and made imputations about Leo's unchastity, implying or asserting homosexuality. Suggestions of homosexual attraction appear in works by two contemporary historians, Francesco Guicciardini and Paolo Giovio. Zimmerman notes Giovio's "disapproval of the pope's familiar banter with his chamberlains – handsome young men from noble families – and the advantage he was said to take of them."

Luther spent a month in Rome in 1510, three years before Leo became pontiff, and was disillusioned at the corruption he found there. In 1520, the year before his excommunication from the Catholic Church, Luther claimed that Leo lived a "blameless life." However, Luther later distanced himself from this claim and alleged in 1531 that Leo had vetoed a measure that cardinals should restrict the number of boys they kept for their pleasure, "otherwise it would have been spread throughout the world how openly and shamelessly the pope and the cardinals in Rome practice sodomy." Against this allegation is the papal bull Supernae dispositionis arbitrio from 1514 which, inter alia, required cardinals to live "... soberly, chastely, and piously, abstaining not only from evil but also from every appearance of evil" and a contemporary and eye-witness at Leo's Court (Matteo Herculaneo), emphasized his belief that Leo was chaste all his life.

Historians have dealt with the issue of Leo's sexuality at least since the late 18th century, and few have given credence to the imputations made against him in his later years and decades following his death, or else have at least regarded them as unworthy of notice; without necessarily reaching conclusions on whether he was homosexual. Those who stand outside this consensus generally fall short of concluding with certainty that Leo was unchaste during his pontificate. Joseph McCabe accused Pastor of untruthfulness and Vaughan of lying in the course of their treatment of the evidence, pointing out that Giovio and Guicciardini seemed to share the belief that Leo engaged in "unnatural vice" (homosexuality) while pope.

Benevolence
Leo X made charitable donations of more than 6,000 ducats annually to retirement homes, hospitals, convents, discharged soldiers, pilgrims, poor students, exiles, cripples, and the sick and unfortunate.

Death
Pope Leo X died suddenly of pneumonia at the age of 45 on 1 December 1521 and was buried in Santa Maria sopra Minerva in Rome. His death came just 10 months after he had excommunicated Martin Luther, the seminal figure in the Protestant Reformation, who was accused of 41 errors in his teachings.

Failure to stem the Reformation
Possibly the most lasting legacy of the reign of Pope Leo X was his perceived failure not just to stem the Reformation but to fuel it. A key issue was that his pontificate failed to bring about the reforms decreed by the Fifth Lateran Council (held between 1512 and 1517) which aimed to deal with many of their political problems as well as to reform Christendom, specifically relating to the papacy, cardinals, and curia. Some believe enforcing these decrees may have been enough to dampen support for radical challenges to church authority. But instead under his leadership, Rome's fiscal and political problems were deepened. A major contributor was his lavish spending (especially on the arts and himself) which led the papal treasury into mounting debt and his decision to authorize the sale of indulgences. The exploitation of people and corruption of religious principles linked to the practice of selling indulgences quickly became the key stimulus for the onset of the Protestant Reformation. Martin Luther's 95 Theses, otherwise entitled "Disputation on the Power and Efficacy of Indulgences", was posted on a Church door in Wittenberg, Germany in October 1517 just seven months after the Lateran V was completed. But Pope Leo X's attempt to prosecute Luther's teaching on indulgences, and to eventually excommunicate him in January 1521, did not get rid of Lutheran doctrine but had the opposite effect of further splintering the Western church.

Excessive spending
Leo was renowned for spending money lavishly on the arts; on charities; on benefices for his friends, relatives, and even people he barely knew; on dynastic wars, such as the War of Urbino; and on his own personal luxury. Within two years of becoming Pope, Leo X spent all of the treasure amassed by the previous Pope, the frugal Julius II, and drove the Papacy into deep debt. By the end of his pontificate in 1521, the papal treasury was 400,000 ducats in debt. This debt contributed not only to the calamities of Leo's own pontificate (particularly the sale of indulgences that precipitated Protestantism) but severely constrained later pontificates (Pope Adrian VI; and Leo's beloved cousin, Clement VII) and forced austerity measures.

Leo X's personal spending was likewise vast. For 1517, his personal income is recorded as 580,000 ducats, of which 420,000 came from the states of the Church, 100,000 from annates, and 60,000 from the composition tax instituted by Sixtus IV. These sums, together with the considerable amounts accruing from indulgences, jubilees, and special fees, vanished as quickly as they were received. To remain financially solvent, the Pope resorted to desperate measures: instructing his cousin, Cardinal Giulio de' Medici, to pawn the Papal jewels; palace furniture; tableware; and even statues of the apostles. Additionally, Leo sold cardinals' hats; memberships to a fraternal order he invented in 1520, the Papal Knights of St. Peter and St. Paul; and borrowed such immense sums from bankers that upon his death, many were ruined.

At Leo's death, the Venetian ambassador Gradenigo estimated the number of the Church's paying offices at 2,150, with a capital value of approximately 3,000,000 ducats and a yearly income of 328,000 ducats.

Patron of learning
Leo X raised the Church to a high rank as the friend of whatever seemed to extend knowledge or to refine and embellish life. He made the capital of Christendom, Rome, a centre of European culture. While yet a cardinal, he had restored the church of Santa Maria in Domnica after Raphael's designs; and as pope he had San Giovanni dei Fiorentini, on the Via Giulia, built, after designs by Jacopo Sansovino and pressed forward the work on St Peter's Basilica and the Vatican under Raphael and Agostino Chigi. Leo's constitution of 5 November 1513 reformed the Roman university, which had been neglected by Julius II. He restored all its faculties, gave larger salaries to the professors, and summoned distinguished teachers from afar; and, although it never attained to the importance of Padua or Bologna, it nevertheless possessed in 1514 a faculty (with a good reputation) of eighty-eight professors.

Leo called Janus Lascaris to Rome to give instruction in Greek, and established a Greek printing press from which the first Greek book printed in Rome appeared in 1515. He made Raphael custodian of the classical antiquities of Rome and the vicinity, the ancient monuments of which formed the subject of a famous letter from Raphael to the pope in 1519. The distinguished Latinists Pietro Bembo and Jacopo Sadoleto were papal secretaries, as well as the famous poet Bernardo Accolti. Other poets, such as Marco Girolamo Vida, Gian Giorgio Trissino and Bibbiena, writers of novelle like Matteo Bandello, and a hundred other literati of the time were bishops, or papal scriptors or abbreviators, or in other papal employs.

"Under his pontificate, Latin Christianity assumed a pagan, Greco-Roman character, which, passing from art into manners, gives to this epoch a strange complexion. Crimes for the moment disappeared, to give place to vices; but to charming vices, vices in good taste, such as those indulged in by Alcibiades and sung by Catullus. —Alexandre Dumas père"

Statesman
Several minor events of Leo's pontificate are worthy of mention. He was particularly friendly with King Manuel I of Portugal as a result of the latter's missionary enterprises in Asia and Africa. Pope Leo X was granted a large embassy from the Portuguese king furnished with goods from Manuel's colonies. His concordat with Florence (1516) guaranteed the free election of the clergy in that city.

His constitution of 1 March 1519 condemned the King of Spain's claim to refuse the publication of papal bulls. He maintained close relations with Poland because of the Turkish advance and the Polish contest with the Teutonic Knights. His bull of July 1519, which regulated the discipline of the Polish Church, was later transformed into a concordat by Clement VII.

Leo showed special favours to the Jews and permitted them to erect a Hebrew printing-press in Rome. Under Daniel Bomberg, that press produced manuscripts of the Talmud and Mikraot Gedolot with Leo's approval and protection.

He approved the formation of the Oratory of Divine Love, a group of pious men in Rome which later became the Theatine Order, and he canonized Francis of Paola.

In popular culture

 * In the 1965 film The Agony and the Ecstasy, Pope Leo is played, while still a Cardinal, by actor Adolfo Celi.

title: Pope Leo X


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Beta particle

A beta particle, also called beta ray or beta radiation (symbol β), is a high-energy, high-speed electron or positron emitted by the radioactive decay of an atomic nucleus during the process of beta decay. There are two forms of beta decay, β− decay and β+ decay, which produce electrons and positrons respectively.

Beta particles with an energy of 0.5 MeV have a range of about one metre in the air; the distance is dependent on the particle energy.

Beta particles are a type of ionizing radiation and for radiation protection purposes are regarded as being more ionising than gamma rays, but less ionising than alpha particles. The higher the ionising effect, the greater the damage to living tissue, but also the lower the penetrating power of the radiation.

β− decay (electron emission)

An unstable atomic nucleus with an excess of neutrons may undergo β− decay, where a neutron is converted into a proton, an electron, and an electron antineutrino (the antiparticle of the neutrino):

This process is mediated by the weak interaction. The neutron turns into a proton through the emission of a virtual W− boson. At the quark level, W− emission turns a down quark into an up quark, turning a neutron (one up quark and two down quarks) into a proton (two up quarks and one down quark). The virtual W− boson then decays into an electron and an antineutrino.

β− decay commonly occurs among the neutron-rich fission byproducts produced in nuclear reactors. Free neutrons also decay via this process. Both of these processes contribute to the copious quantities of beta rays and electron antineutrinos produced by fission-reactor fuel rods.

β+ decay (positron emission)
Unstable atomic nuclei with an excess of protons may undergo β+ decay, also called positron decay, where a proton is converted into a neutron, a positron, and an electron neutrino:

Beta-plus decay can only happen inside nuclei when the absolute value of the binding energy of the daughter nucleus is greater than that of the parent nucleus, i.e., the daughter nucleus is a lower-energy state.

Beta decay schemes
The accompanying decay scheme diagram shows the beta decay of caesium-137. 137Cs is noted for a characteristic gamma peak at 661 KeV, but this is actually emitted by the daughter radionuclide 137mBa. The diagram shows the type and energy of the emitted radiation, its relative abundance, and the daughter nuclides after decay.

Phosphorus-32 is a beta emitter widely used in medicine and has a short half-life of 14.29 days and decays into sulfur-32 by beta decay as shown in this nuclear equation:
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1.709 MeV of energy is released during the decay. The kinetic energy of the electron varies with an average of approximately 0.5 MeV and the remainder of the energy is carried by the nearly undetectable electron antineutrino. In comparison to other beta radiation-emitting nuclides, the electron is moderately energetic. It is blocked by around 1 m of air or 5 mm of acrylic glass.
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Interaction with other matter
Of the three common types of radiation given off by radioactive materials, alpha, beta and gamma, beta has the medium penetrating power and the medium ionising power. Although the beta particles given off by different radioactive materials vary in energy, most beta particles can be stopped by a few millimeters of aluminium. However, this does not mean that beta-emitting isotopes can be completely shielded by such thin shields: as they decelerate in matter, beta electrons emit secondary gamma rays, which are more penetrating than betas per se. Shielding composed of materials with lower atomic weight generates gammas with lower energy, making such shields somewhat more effective per unit mass than ones made of high-Z materials such as lead.

Being composed of charged particles, beta radiation is more strongly ionizing than gamma radiation. When passing through matter, a beta particle is decelerated by electromagnetic interactions and may give off bremsstrahlung x-rays.

In water, beta radiation from many nuclear fission products typically exceeds the speed of light in that material (which is 75% that of light in vacuum), and thus generates blue Cherenkov radiation when it passes through water. The intense beta radiation from the fuel rods of swimming pool reactors can thus be visualized through the transparent water that covers and shields the reactor (see illustration at right).

Detection and measurement
The ionizing or excitation effects of beta particles on matter are the fundamental processes by which radiometric detection instruments detect and measure beta radiation. The ionization of gas is used in ion chambers and Geiger–Müller counters, and the excitation of scintillators is used in scintillation counters. The following table shows radiation quantities in SI and non-SI units:

 * The gray (Gy), is the SI unit of absorbed dose, which is the amount of radiation energy deposited in the irradiated material. For beta radiation this is numerically equal to the equivalent dose measured by the sievert, which indicates the stochastic biological effect of low levels of radiation on human tissue. The radiation weighting conversion factor from absorbed dose to equivalent dose is 1 for beta, whereas alpha particles have a factor of 20, reflecting their greater ionising effect on tissue.
 * The rad is the deprecated CGS unit for absorbed dose and the rem is the deprecated CGS unit of equivalent dose, used mainly in the USA.

Beta spectroscopy
The energy contained within individual beta particles is measured via beta spectrometry; the study of the obtained distribution of energies as a spectrum is beta spectroscopy. Determination of this energy is done by measuring the amount of deflection of the electron's path under a magnetic field.

Applications
Beta particles can be used to treat health conditions such as eye and bone cancer and are also used as tracers. Strontium-90 is the material most commonly used to produce beta particles.

Beta particles are also used in quality control to test the thickness of an item, such as paper, coming through a system of rollers. Some of the beta radiation is absorbed while passing through the product. If the product is made too thick or thin, a correspondingly different amount of radiation will be absorbed. A computer program monitoring the quality of the manufactured paper will then move the rollers to change the thickness of the final product.

An illumination device called a betalight contains tritium and a phosphor. As tritium decays, it emits beta particles; these strike the phosphor, causing the phosphor to give off photons, much like the cathode-ray tube in a television. The illumination requires no external power, and will continue as long as the tritium exists (and the phosphors do not themselves chemically change); the amount of light produced will drop to half its original value in 12.32 years, the half-life of tritium.

Beta-plus (or positron) decay of a radioactive tracer isotope is the source of the positrons used in positron emission tomography (PET scan).

History
Henri Becquerel, while experimenting with fluorescence, accidentally found out that uranium exposed a photographic plate, wrapped with black paper, with some unknown radiation that could not be turned off like X-rays.

Ernest Rutherford continued these experiments and discovered two different kinds of radiation:

 * alpha particles that did not show up on the Becquerel plates because they were easily absorbed by the black wrapping paper
 * beta particles which are 100 times more penetrating than alpha particles.

He published his results in 1899.

In 1900, Becquerel measured the mass-to-charge ratio ($m/e$) for beta particles by the method of J. J. Thomson used to study cathode rays and identify the electron. He found that $e/m$ for a beta particle is the same as for Thomson's electron, and therefore suggested that the beta particle is in fact an electron.

Health
Beta particles are moderately penetrating in living tissue, and can cause spontaneous mutation in DNA.

Beta sources can be used in radiation therapy to kill cancer cells.

title: Beta particle


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August Ferdinand Möbius

August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer.

Early life and education
Möbius was born in Schulpforta, Electorate of Saxony, and was descended on his mother's side from religious reformer Martin Luther. He was home-schooled until he was 13, when he attended the college in Schulpforta in 1803, and studied there, graduating in 1809. He then enrolled at the University of Leipzig, where he studied astronomy under the mathematician and astronomer Karl Mollweide. In 1813, he began to study astronomy under mathematician Carl Friedrich Gauss at the University of Göttingen, while Gauss was the director of the Göttingen Observatory. From there, he went to study with Carl Gauss's instructor, Johann Pfaff, at the University of Halle, where he completed his doctoral thesis The occultation of fixed stars in 1815. In 1816, he was appointed as Extraordinary Professor to the "chair of astronomy and higher mechanics" at the University of Leipzig. Möbius died in Leipzig in 1868 at the age of 77. His son Theodor was a noted philologist.

Contributions
He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict Listing a few months earlier. The Möbius configuration, formed by two mutually inscribed tetrahedra, is also named after him. Möbius was the first to introduce homogeneous coordinates into projective geometry. He is recognized for the introduction of the Barycentric coordinate system. Before 1853 and Schläfli's discovery of the 4-polytopes, Möbius (with Cayley and Grassmann) was one of only three other people who had also conceived of the possibility of geometry in more than three dimensions.

Many mathematical concepts are named after him, including the Möbius plane, the Möbius transformations, important in projective geometry, and the Möbius transform of number theory. His interest in number theory led to the important Möbius function μ(n) and the Möbius inversion formula. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.

Collected works

 * Gesammelte Werke erster Band (v. 1)  (Leipzig : S. Hirzel, 1885)
 * Gesammelte Werke zweiter Band (v. 2)  (Leipzig : S. Hirzel, 1885)
 * Gesammelte Werke dritter Band (v. 3)  (Leipzig : S. Hirzel, 1885)
 * Gesammelte Werke vierter Band (v. 4)  (Leipzig : S. Hirzel, 1885)
 * Die elemente der mechanik des himmels, auf neuem wege ohne hülfe höherer rechnungsarten dargestellt von August Ferdinand Möbius (Leipzig, Weidmann'sche buchhandlung, 1843)

title: August Ferdinand Möbius