Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.

Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original.

Although the play does show a certain structural amateurishness ( there are eleven acts varying in length from twenty-five seconds to an hour and a half ), the statement it makes concerning the ceaseless yearning and searching of youth is profound and worthy of our attention.

Although Pearson disbelieved almost everything Lizzie said, and read a sinister purpose into almost everything she did, he happily accepted her statement about Bridget as the whole truth.

Although the project had an excellent pedigree with stars Spencer Tracy and Katharine Hepburn, the film was not a success, and Capra's eyebrow-raising statement, " I think State of the Union was my most perfect film in handling people and ideas " has few adherents today.

Although the statement of the theory in terms of graphs may imply perturbation theory, use of graphical methods in the many-body problem shows that this formalism is flexible enough to deal with phenomena of nonperturbative characters ...

Although the scale statement is nominal it is usually accurate enough for all but the most precise of measurements.

Although the police likely questioned Page, no written account of her statement has surfaced.

Although Paine liked to say that he read very little, his writings belie this statement ; The Age of Reason has intellectual roots in the traditions of David Hume, Spinoza, and Voltaire.

Although Watson is well known for his alleged 1943 statement, " I think there is a world market for maybe five computers ", there is scant evidence he made it.

Although commonly thought of as a unified statement of doctrine by the two reformers, Luther did not conceal his dissatisfaction with the irenic tone of the confession.

Although most mathematicians and physicists ( and many philosophers ) would accept the statement " mathematics is a language ", linguists believe that the implications of such a statement must be considered.

" Although Peres himself did not retract the statement, the Israeli Foreign Ministry later issued a cable to its missions which stated that the " The minister absolutely did not say, as the Turkish news agency alleged, “ What the Armenians underwent was a tragedy, not a genocide "".

Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors.

Although this statement could be read as making a procedural point ( that the Commission must involve people of diverse cultures, especially cultures that had been or are still under European colonial or imperial domination ), the document ended by making two substantive claims:

Although Article VIII itself states that the three Catholic creeds are a sufficient statement of faith, the Articles have often been perceived as the nearest thing to a supplementary confession of faith possessed by the tradition.

Although the statute wasn't specific about whether it was legal for a defendant to read his statement, Spicer allowed his testimony to proceed.

Although Ermakov's archived statement does not say so, he told his wife that Anastasia was finished off with bayonets, while Yurovsky wrote that as the bodies were carried out, one or more of the girls cried out and were clubbed on the back of the head.

" The statement continued, " Although Mr. Fields's intent is clearly to slander and discredit Dr. Drew, under no circumstances is Dr. Drew making a blanket diagnosis about Scientology nor Mr. Cruise whom he does not know.

Although the element chob has been translated as ' squint-eyed ', which is an iconographic feature of the Classic sun deity, the only source for this translation Is a single statement by Tozzer.

Although the Proclamation presents no new doctrines or policies, it provides an official statement of the church on gender and sex.

Although the 1962 novel pre-dated extensive US activity in Vietnam and was not based on any historic incidents, by the December 1966 release of the film it was seen as an explicit statement on the US's extensive combat involvement in the Vietnam War in reviews published by the New York Times.

Although he did issue a statement regretting the " appearance " and " impression " of the letters he wrote on government stationery, he maintained that he " never sought to leverage my public position nor mix my government role with my personal and professional relationship " in the letters.

Although the fragmentary nature of his music militates against classical thematic unity, he often employs structural blocks of sound, where similar rhythms and thematic material allude to previous passages ( as opposed to classical statement and recapitulation ).

Although Alaska Sen. Ted Stevens felt the statement " was not written by a proponent ," it maintained the general approval for pipeline construction that was demonstrated in the draft statement.

Although these postulates cannot be applied in all cases, they do retain historical importance to the development of scientific thought and are still being used today.

Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit.

Although the model was based mainly on the characterization of United States media, Chomsky and Herman believe the theory is equally applicable to any country that shares the basic economic structure and organizing principles which the model postulates as the cause of media biases.

Although the story postulates a race of intelligent non-humans, it is set in the Foundation universe, during the era of Trantor's Galactic Empire.

Although it was dark as usual I could see that the hall had only recently contained a great many people.

Although there were only four fluids tested, it was apparent that there were two distinct types.

Although biopsies have shown structural changes in some of the reported cases of steroid-induced weakness, this case provides the only example known to us in which necropsy afforded the opportunity for extensive study of multiple muscle groups.

Although she weighed only 108 pounds when she visited him, Carroll permitted her to go on a 10-day fast in which she took nothing but water.

Although the training is only for one month, it is intensive and thorough.

Although the primary mathematical properties of the middle number at the center of the Lo Shu, and the interrelation of all the other numbers to it, might seem enough to account for the deep fascination which the Lo Shu held for the Old Chinese philosophers, this was actually only a beginning of wonders.

Although it was only three o'clock, he stopped in at the Golden Calf.

Although Lincoln won only a plurality of the popular vote, his victory in the electoral college was decisive: Lincoln had 180 and his opponents added together had only 123.

Although the name actually applies to a group of snakes, it is often used to refer only to one species in particular, the common or green anaconda, Eunectes murinus, which is one of the largest snakes in the world.

Although the entrance to the chamber warns against disturbing him, Shrewsbury opens Alhazred's sarcophagus anyway, finding that only rags, bones, and dust remain of Alhazred.

Although only one of his four children survived to adulthood, Antoninus came to be ancestor to generations of prominent Roman statesmen and socialites, including at least one empress consort.

Although at least three pads were planned, only two, designated A and B, were completed in October 1965.

Although amaranth was cultivated on a large scale in ancient Mexico, Guatemala, and Peru, nowadays it is only cultivated on a small scale there, along with India, China, Nepal, and other tropical countries ; thus, there is potential for further cultivation in those countries, as well as in the U. S. In a 1977 article in Science, amaranth was described as " the crop of the future.

Although the Visigoths plundered Rome, they treated its inhabitants humanely and burned only a few buildings.

Although many other chemists have repeated this advice, IUPAC and most chemistry texts still favour the usage of allotrope and allotropy for elements only.

Although Ares received occasional sacrifice from armies going to war, the god had a formal temple and cult at only a few sites.

Although used for similar purposes, transparent colors and image masks do not permit the smooth blending of the superimposed image pixels with those of the background ( only whole image pixels or whole background pixels allowed ).

Although cheap and effective in enabling use of some software that only used official ROM entry points for text output, this solution proved very slow because the Electron had to be placed into an 80-byte-pitch display to be able to get anywhere near to reproducing mode 7 and the CPU spent a lot of time drawing approximations of mode 7 characters and graphics that in a hardware solution would be achieved without any CPU processing.

Although this produced a mode 7 that barely impacted upon CPU performance and gave the same visual quality as the BBC Micro, it remained compatible only with software that used the ROM routines for outputting text and graphics and still used 10 kB of memory for the display.

Although programs can alter the position of the screen in memory, the non-linear format of the display means that vertical scrolling can only be done in blocks of 8 pixels without further work.

Although there have been periodic initiatives to revive their use, airships have seen only niche application since that time.

Although most audio file formats support only one type of audio data ( created with an audio coder ), a multimedia container format ( as Matroska or AVI ) may support multiple types of audio and video data.

Although the absolute numerical values of astronomical and historical years only differ by one before year 1, this difference is critical when calculating astronomical events like eclipses or planetary conjunctions to determine when historical events which mention them occurred.

Although it has only 6. 14 percent of Alaska's land area, it is larger than the state of Maine, and almost as large as the state of Indiana.