Although modern scholars have expressed surprise that `` the simple magic square of three '', a mere `` mathematical puzzle '', was able to exert a considerable influence on the minds and imaginations of the cultured Chinese for so many centuries, they could have found most of the answers right within the square itself.

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.

Although the idea expressed in Cogito ergo sum is widely attributed to Descartes, he was not the first to mention it.

Although artists working with other media have expressed interest in the concept, such as Roy Ascott, " cyberspace " in digital art is mostly used as a synonym for immersive virtual reality and remains more discussed than enacted.

Although the shooter's motives remain unknown, relatives of the shooter had expressed concerns about his mental health.

Although the Casimir effect can be expressed in terms of virtual particles interacting with the objects, it is best described and more easily calculated in terms of the zero-point energy of a quantized field in the intervening space between the objects.

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests.

" Although not expressed as strongly, Glyn Jones believed that he and Thomas ' friendship cooled in the later years as he had not ' rejected enough ' of the elements that Thomas disliked – " Welsh nationalism and a sort of hill farm morality ".

Although most obituaries were positive, a number of clergymen expressed different sentiments.

Although lightbulb jokes tend to be derogatory in tone, the people targeted by them may take pride in the stereotypes expressed and are often themselves the jokes ' originators.

Although color gamut can be expressed as three pairs of numbers, the XY coordinates within color space of the reddest red, greenest green, and bluest blue, it is usually expressed as a ratio of the total area within color space that a display can show relative to some standard such as saying that a display was " 120 % of NTSC ".

Although originally expressed in Newton's third law, the conservation of linear momentum also holds in special relativity ( with a modified formula ) and, with appropriate definitions, a ( generalized ) linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory, and general relativity.

Although Germany's once-powerful trade unions distanced themselves from the Left in the 2005 election, some union leaders expressed interest in cooperating with the party after the election.

Although he has expressed admiration for many of the values promoted by secular humanism, Singer believes it to be incomplete and promotes a preference utilitarian view he calls " personism " instead.

Although he approved of the doctrine thus expressed, he also ordered that the Nicene Creed, without Filioque, be displayed on silver tablets placed in Saint Peter's Basilica, adding: " Haec Leo posui amore et cautela orthodoxae fidei " ( I, Leo, put these here for love and protection of orthodox faith ).

Although constitutively expressed proteasomes can participate in this process, a specialized complex composed of proteins whose expression is induced by interferon gamma produces peptides of the optimal size and composition for MHC binding.

Although both the exclusivist and the pluralist may agree on the existence of religious or spiritual reality, the pluralist recognizes that this reality is expressed in different cultures and by different people in different ways.

Although stellar parameters can be expressed in SI units or CGS units, it is often most convenient to express mass, luminosity, and radii in solar units, based on the characteristics of the Sun:

Although she expressed a dislike for singing in those bars, Twain believes that this was her own kind of performing arts school on the road.

Although expressed as a problem of software design, the second-system effect is observable throughout all human design effort.

Although the goal is to improve trade, the president of Tajikistan expressed concern that it could also increase drug traffic.

Although they can be written in a completely symbolic form using, for example, propositional calculus, theorems are often expressed in a natural language such as English.

Although the hyperfilament is constructed from " continuous pseudo-one-dimensional diamond crystal ", Clarke later expressed his belief that another type of carbon, Buckminsterfullerene, would play the role of hyperfilament in a real space elevator.

Although a straight drum shade would be adequate and sufficiently neutral that the puppets could be changed without disharmony, it is far more fun to create shades in the gay spirit of a child's playtime.

Although both concede they would like to hit 60, they stick primarily to the baseball player's standard quote: `` The important thing is to win the pennant ''.

Although he was not an innovator, he would not follow the absolute letter of the law ; rather he was driven by concerns over humanity and equality, and introduced into Roman law many important new principles based upon this notion.

Although Albert has received relatively little recognition in German history, his dissolution of the Teutonic State caused the founding of the Duchy of Prussia ( and also the Hohenzollern dynasty ), which would eventually become arguably the most powerful German state and instrumental in uniting the whole of Germany.

Although there is no cliff at the site where his body was found there is a very steep rocky embankment-which would have been fatal in the dark.

Although rats would attack him at night, he was able, by domesticating and living near feral cats, to sleep soundly and in safety.

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 the Carthaginians conquered much of the land around Alicante, the Romans would eventually rule Hispania Tarraconensis for over 700 years.

Although this is unlikely ( as Calraet would likely have signed his paintings “ A.

Although the Atari 5200's internal design was extensively based on that of the 400 / 800 home computers, the differences were sufficient that games designed for one would not run directly on the other.

Although Capp's endorsement activities never rivaled Li ' l Abner's or Fearless Fosdick's, he was a celebrity spokesman in print ads for Sheaffer Snorkel fountain pens ( along with colleagues and close friends Milton Caniff and Walt Kelly ), and — with an irony that would become apparent later — a brand of cigarettes, ( Chesterfield ).

Although it has been known for a while that the Black Sea and the Sea of Marmara flow into each other in an example of a density flow, findings of a study by the University of Leeds in August 2010 reveal that there is in fact an underwater channel of high density water flowing across the floor of the Bosphorus ( caused by the difference in density of the two seas ), which would be the sixth largest river on Earth if it were to be on land.

Although Henry Lumley ’ s British cavalry had managed to cross the marshy ground around the Petite Gheete, it was soon evident to Marlborough that sufficient cavalry support would not be practicable and that the battle could not be won on the Allied right.

Although he would go on to have back-to-back 17-win seasons for the Chicago Cubs in 1971 and 1972, including a no-hitter in the latter season, this did not help the Reds, who ended up losing the 1970 World Series to Robinson and the Orioles.

Although Williams would hit 63 home runs in two seasons with Atlanta, he would only hit 36 homers in two seasons with the Orioles.

Although he was aware that the Bastarnae were hardly likely to achieve the same success as Hannibal some 40 years earlier, and would most likely end up cut to pieces by the Romans, Philip hoped that the Romans would be distracted long enough to allow him to reoccupy his former possessions in Greece.

Although bongos are quite easy for humans to catch via snares ; it is of interest that many people native to the bongos habitat believed that if they ate or touched bongo they would have spasms similar to epileptic seizures.

Although production of Luftwaffe fighter aircraft continued, they would be unable to fly for lack of fuel.

Although Pasternak never left his wife, this initiated an extramarital relationship which would last for the remainder of Pasternak's life.

Although there were many IRA men — both Official and Provisional — present at the protest, it is claimed they were all unarmed, apparently because it was anticipated that the paratroopers would attempt to " draw them out ".

Although the film would have comic moments, Chaplin described it as first and foremost a romantic film.

Although Love said she would " never write a book ", she did publish a memoir in 2006 titled Dirty Blonde: The Diaries of Courtney Love.