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 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 counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description.

Although recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave Coriolis, in connection with the theory of water wheels, Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.

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 it is a mathematical abstraction rather than a programming language, it forms the basis of almost all functional programming languages today.

Although this model is conceptually helpful, its mathematical formulation makes several assumptions that limit its applicability to real systems.

Although the existence of molecules has been accepted by many chemists since the early 19th century as a result of Dalton's laws of Definite and Multiple Proportions ( 1803 – 1808 ) and Avogadro's law ( 1811 ), there was some resistance among positivists and physicists such as Mach, Boltzmann, Maxwell, and Gibbs, who saw molecules merely as convenient mathematical constructs.

Although Escher did not have mathematical training — his understanding of mathematics was largely visual and intuitive — Escher's work had a strong mathematical component, and more than a few of the worlds which he drew are built around impossible objects such as the Necker cube and the Penrose triangle.

Although simple programs can be written in a few hours, programs that use complex mathematical formulas whose solutions can only be approximated or that draw data from many existing systems may require more than a year of work.

Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of argument as universally valid.

Although it is normal to regard the quantity demanded and the quantity supplied as functions of the price of the good, the standard graphical representation, usually attributed to Alfred Marshall, has price on the vertical axis and quantity on the horizontal axis, the opposite of the standard convention for the representation of a mathematical function.

Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter, and introduced new mathematical classes of objects that are now essential elements of fundamental physics.

Although the development of mathematical logic had made applications to algebra possible, they came about slowly ; results published by Anatoly Maltsev in the 1940s went unnoticed because of the war.

Although self-organised criticality has been able to reproduce 1 / f noise in sandpile models, there are no simple mathematical models to create pink noise.

Although the social theories and quasi-empiricism, and especially the embodied mind theory, have focused more attention on the epistemology implied by current mathematical practices, they fall far short of actually relating this to ordinary human perception and everyday understandings of knowledge.

Although Strabo cited the antique Greek astronomers Eratosthenes and Hipparchus, acknowledging their astronomical and mathematical efforts towards geography, he claimed that a descriptive approach was more practical, such that his works were designed for statesmen who were more anthropologically than numerically concerned with the character of countries and regions.

Sir Michael Atiyah said of Witten, " Although he is definitely a physicist, his command of mathematics is rivaled by few mathematicians ... Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems ... he has made a profound impact on contemporary mathematics.

Although mathematics was not part of his course work, de Moivre read several mathematical works on his own including Elements de mathematiques by Father Prestet and a short treatise on games of chance, De Ratiociniis in Ludo Aleae, by Christiaan Huygens.

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids.

Although to date, no single theory has successfully expressed these relationships in viable mathematical or physical terms.

Although originally considered a subspecies of A. ramidus, in 2004 anthropologists Yohannes Haile-Selassie, Gen Suwa, and Tim D. White published an article elevating A. kadabba to species level on the basis of newly-discovered teeth from Ethiopia.

Although much of the church hierarchy in the East had opposed the Nicene creed in the decades leading up to Theodosius ' accession, he managed to achieve unity on the basis of the Nicene creed.

Although this shift was an important one, it did not represent a radical break from the past so much as a small step in a broader, more gradual socio-economic movement that had been going on at least since 1907 when van de Velde had argued for a craft basis for design while Hermann Muthesius had begun implementing industrial prototypes.

Although traditionally such cladograms were generated largely on the basis of morphological characters and originally calculated by hand, genetic sequencing data and computational phylogenetics are now very commonly used in phylogenetic analysis.

Although the economy operates on a free-market basis, the lack of private and government investment, shortages of skilled labor and experienced managers, and inflation and the devaluation of the Turkish lira continue to plague the economy.

Although Diffie – Hellman key agreement itself is an anonymous ( non-authenticated ) key-agreement protocol, it provides the basis for a variety of authenticated protocols, and is used to provide perfect forward secrecy in Transport Layer Security's ephemeral modes ( referred to as EDH or DHE depending on the cipher suite ).

Although most of the relevant tissues and endocrine glands had been identified by early anatomists, a more humoral approach to understanding biological function and disease was favoured by the ancient Greek and Roman thinkers such as Aristotle, Hippocrates, Lucretius, Celsus, and Galen, according to Freeman et al., and these theories held sway until the advent of germ theory, physiology, and organ basis of pathology in the 19th century.

Although mythical elements are not so prominent in Exodus as in Genesis, the echoes of ancient legends are crucial to understanding the book's origins and purpose: for example, the story of the infant Moses's salvation from the Nile has its basis in an earlier legend of king Sargon, while the story of the parting of the Red Sea trades on Mesopotamian creation mythology.

Although this is an independent definition that does not refer to the older basis of ephemeris time, it uses the same quantity as the value of the ephemeris second measured by the cesium clock in 1958.

Although experts are legally prohibited from expressing their opinion of submitted evidence until after they are hired, sometimes a party can surmise beforehand, because of reputation or prior cases, that the testimony will be favorable regardless of any basis in the submitted data ; such experts are commonly disparaged as " hired guns.

Although he did not know the physical basis for heredity, Mendel observed that organisms inherit traits via discrete units of inheritance, which are now called genes.

Although Haddad notes that the authenticity of both hadeeth is questionable, he nevertheless concludes that the underlying principle of superiority internal jihad does have a reliable basis in the Quran and other writings.

Although the Pancha Sila ( Five Principles of Peaceful Coexistence ) was the basis of the 1954 Sino-Indian border treaty, in later years, Nehru's foreign policy suffered through increasing Chinese assertiveness over border disputes and Nehru's decision to grant political asylum to the 14th Dalai Lama.

Although several historical figures have been proposed as the basis for Arthur, no convincing evidence for these identifications has emerged.

Although the legal force of the document was not established and it was not incorporated directly into either the American or German law, the Nuremberg Code and the related Declaration of Helsinki are the basis for the Code of Federal Regulations Title 45 Volume 46, which are the regulations issued by the United States Department of Health and Human Services governing federally-funded human subjects research in the United States.

Although professions enjoy high status and public prestige, not all professionals earn high salaries, and even within specific professions there exist significant inequalities of compensation ; for example, a trial lawyer specializing in tort litigation on a contingent-fee basis may earn several times what a prosecutor or public defender earns.

Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred.

Although there is much debate as to which states or groups operate strictly according to Islamic Law, Sharia is the official basis for state laws in the following countries: Yemen, Afghanistan, Somalia, Sudan, Saudi Arabia, Mauritania, Oman and Iran.

Although on the basis of history and precedent the Sixth Amendment mandates unanimity in a federal jury trial, the Supreme Court has ruled that the Due Process Clause of the Fourteenth Amendment, while requiring States to provide jury trials for serious crimes, does not incorporate all the elements of a jury trial within the meaning of the Sixth Amendment and does not require jury unanimity.

Although the zodiac remains the basis of the ecliptic coordinate system in use in astronomy besides the equatorial one, the term and the names of the twelve signs are today mostly associated with horoscopic astrology.

Although research suggests the likelihood of a genetic basis, there is no known genetic etiology and brain imaging techniques have not identified a clear common pathology.

Although FDDI logical topology is a ring-based token network, it does not use the IEEE 802. 5 token ring protocol as its basis ; instead, its protocol is derived from the IEEE 802. 4 token bus timed token protocol.