Euclid's original proof adds a third: the two lengths are not prime to one another.

Some folk etymologies argue that the original forms of these symbols indicated their value through the number of angles they contained, however there is no proof of any such origin.

Suggested locations for backups are a fireproof, waterproof, and heat proof safe, or in a separate, offsite location than that in which the original files are contained.

Investors in the VOC were issued paper certificates as proof of share ownership, and were able to trade their shares on the original Amsterdam stock exchange.

The original versions of his papers contained " many technical errors of varying degree "; when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception — a claimed proof of the Continuum hypothesis.

The original proof is based on the Taylor series expansions of the exponential function e < sup > z </ sup > ( where z is a complex number ) and of sin x and cos x for real numbers x ( see below ).

Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics.

Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument.

In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's original proof.

While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, and later proofs have introduced various simplifications through the use of Tauberian theorems but remained difficult to digest, a surprisingly short proof was discovered in 1980 by American mathematician Donald J. Newman.

They initially implied the proof was their own achievement based on the " Hamilton-Perelman theory ", but later retracted the original version of their paper, and posted a revised version, in which they referred to their work as the more modest " exposition of Hamilton – Perelman's proof ".

The original completeness proof applies to all classical models, not some special proper subclass of intended ones.

We will now follow the strategy of David Hilbert ( 1862 – 1943 ) who gave a simplification of the original proof of Charles Hermite.

Where a title has been enacted into positive law, however, a court may neither permit nor require proof of the underlying original statutes.

The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.

His original proof relied partly on Hamilton's work on the Ricci flow.

In his work on the divine proportion, H. E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art — the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor.

He also was the original author of rzip, which uses a similar algorithm to rsync.

The German metal band van Canto uses vocal noises to imitate guitars on covers of well-known rock and metal songs ( such as " Master of Puppets " by Metallica ) as well as original compositions.

Iterated product ciphers carry out encryption in multiple rounds, each which uses a different subkey derived from the original key.

Citi Field now uses a new apple, one that is nearly four times the size of the original.

More modern church buildings have a variety of architectural styles and layouts ; many buildings that were designed for other purposes have now been converted for church use ; and, similarly, many original church buildings have been put to other uses.

Kosher for Passover Coca-Cola sold in the U. S. around the Jewish holiday also uses sucrose rather than HFCS and is also highly sought after by people who prefer the original taste.

The simplest, and original, implementation of the protocol uses the multiplicative group of integers modulo p, where p is prime and g is primitive root mod p. Here is an example of the protocol, with non-secret values in < span style =" color: blue "> blue </ span >, and secret values in < span style =" color: red "> boldface red </ span >:

Blending uses an implement to soften or spread the original drawing strokes.

It retains the original name and uses Lobotomy's SlaveDriver engine.

) The original Drake Equation can be extended to a more realistic model, where the equation uses not the number of stars that are forming now, but those that were forming several billion years ago.

Davros emerges as " The Great Healer " of the funeral and cryogenic preservation centre Tranquil Repose on the planet Necros in the Sixth Doctor story Revelation of the Daleks, where he uses frozen bodies to engineer a new variety of Daleks loyal to him, distinguished from the original Daleks by their white and gold livery and slightly changed design.

George B. Dyson uses the heading of Butler's original article in Darwin Among the Machines: The Evolution of Global Intelligence ( 1998 ) ISBN 0-7382-0030-1.

The original form of electron microscope, the transmission electron microscope ( TEM ) uses a high voltage electron beam to create an image.

This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

One reason for this given by the researchers is that the B-80 is actually more like the original geodesic dome structure popularized by Buckminster Fuller, which uses triangles rather than hexagons.

National flags are potent patriotic symbols with varied wide-ranging interpretations, often including strong military associations due to their original and ongoing military uses.

In the historical description of gazpacho it was remarked that the original recipe uses bread, water, vinegar, oil and salt.

The 802. 11a standard uses the same data link layer protocol and frame format as the original standard, but an OFDM based air interface ( physical layer ).

802. 11b has a maximum raw data rate of 11 Mbit / s and uses the same media access method defined in the original standard.

Originally constructed to fill the capacity of a library, museum, and chapel, its original uses are engraved in the exterior stonework on the east side.

Despite the intent of the original allocation, in recent years the fastest-growing uses of these bands have been for short-range, low power communications systems.

His theory of justice uses a device called the original position to ask us which principles of justice we would choose to regulate the basic institutions of our society if we were behind a ' veil of ignorance.

It also uses the same save system as the Famicom Disk System version does, as opposed to the Password system from the original NES version.

Serilith are mixed media original prints created in a process in which an artist uses the lithograph and serigraph process.

In the original short story, a bald scientist uses a formula created from a piece of alien meteor to give a vagrant named Bill Dunn telepathic abilities, which Dunn abuses for personal gain.