In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light propagates through the material. It is defined as n = c v, where c is the speed of light in vacuum and v is the phase velocity of light in the medium. For example, the refractive index of water is 1.333, meaning that light travels 1.333 times as fast in vacuum as in water. The refractive index determines how much the path of light is bent, or refracted, when entering a material; this is described by Snell's law of refraction, n1 sinθ1 = n2 sinθ2, where θ1 and θ2 are the angles of incidence and refraction of a ray crossing the interface between two media with refractive indices n1 and n2. The refractive indices determine the amount of light, reflected when reaching the interface, as well as the critical angle for total internal reflection and Brewster's angle; the refractive index can be seen as the factor by which the speed and the wavelength of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is v = c/n, the wavelength in that medium is λ = λ0/n, where λ0 is the wavelength of that light in vacuum.
This implies that vacuum has a refractive index of 1, that the frequency of the wave is not affected by the refractive index. As a result, the energy of the photon, therefore the perceived color of the refracted light to a human eye which depends on photon energy, is not affected by the refraction or the refractive index of the medium. While the refractive index affects wavelength, it depends on photon frequency and energy so the resulting difference in the bending angle causes white light to split into its constituent colors; this is called dispersion. It can be observed in prisms and rainbows, chromatic aberration in lenses. Light propagation in absorbing materials can be described using a complex-valued refractive index; the imaginary part handles the attenuation, while the real part accounts for refraction. The concept of refractive index applies within the full electromagnetic spectrum, from X-rays to radio waves, it can be applied to wave phenomena such as sound. In this case the speed of sound is used instead of that of light, a reference medium other than vacuum must be chosen.
The refractive index n of an optical medium is defined as the ratio of the speed of light in vacuum, c = 299792458 m/s, the phase velocity v of light in the medium, n = c v. The phase velocity is the speed at which the crests or the phase of the wave moves, which may be different from the group velocity, the speed at which the pulse of light or the envelope of the wave moves; the definition above is sometimes referred to as the absolute refractive index or the absolute index of refraction to distinguish it from definitions where the speed of light in other reference media than vacuum is used. Air at a standardized pressure and temperature has been common as a reference medium. Thomas Young was the person who first used, invented, the name "index of refraction", in 1807. At the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers; the ratio had the disadvantage of different appearances. Newton, who called it the "proportion of the sines of incidence and refraction", wrote it as a ratio of two numbers, like "529 to 396".
Hauksbee, who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9". Hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1. Young did not use a symbol for the index of refraction, in 1807. In the next years, others started using different symbols: n, m, µ; the symbol n prevailed. For visible light most transparent media have refractive indices between 1 and 2. A few examples are given in the adjacent table; these values are measured at the yellow doublet D-line of sodium, with a wavelength of 589 nanometers, as is conventionally done. Gases at atmospheric pressure have refractive indices close to 1 because of their low density. All solids and liquids have refractive indices above 1.3, with aerogel as the clear exception. Aerogel is a low density solid that can be produced with refractive index in the range from 1.002 to 1.265. Moissanite lies at the other end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, but some high-refractive-index polymers can have values as high as 1.76.
For infrared light refractive indices can be higher. Germanium is transparent in the wavelength region from 2 to 14 µm and has a refractive index of about 4. A type of new materials, called topological insulator, was found holding higher refractive index of up to 6 in near to mid infrared frequency range. Moreover, topological insulator material are transparent; these excellent properties make them a type of significant materials for infrared optics. According to the theory of relativity, no information can travel faster than the speed of light in vacuum, but this does not mean that the refractive index cannot be lower than 1; the refractive index measures the phase velocity of light. The phase velocity is the speed at which the crests of the wave move and can be faster than the speed of light in vacuum, thereby give a refractive index below 1; this can occur close to resonance frequencies, for absorbing media, in plasmas, for X-rays. In the X-ray regime the refractive indices are
Baghdad is the capital of Iraq. The population of Baghdad, as of 2016, is 8,765,000, making it the largest city in Iraq, the second largest city in the Arab world, the second largest city in Western Asia. Located along the Tigris River, the city was founded in the 8th century and became the capital of the Abbasid Caliphate. Within a short time of its inception, Baghdad evolved into a significant cultural and intellectual center for the Islamic world. This, in addition to housing several key academic institutions, as well as hosting multiethnic and multireligious environment, garnered the city a worldwide reputation as the "Centre of Learning". Baghdad was the largest city of the Middle Ages for much of the Abbasid era, peaking at a population of more than a million; the city was destroyed at the hands of the Mongol Empire in 1258, resulting in a decline that would linger through many centuries due to frequent plagues and multiple successive empires. With the recognition of Iraq as an independent state in 1938, Baghdad regained some of its former prominence as a significant center of Arab culture.
In contemporary times, the city has faced severe infrastructural damage, most due to the 2003 invasion of Iraq, the subsequent Iraq War that lasted until December 2011. In recent years, the city has been subjected to insurgency attacks; the war had resulted in a substantial loss of historical artifacts as well. As of 2018, Baghdad was listed as one of the least hospitable places in the world to live, ranked by Mercer as the worst of 231 major cities as measured by quality-of-life; the name Baghdad is pre-Islamic, its origin is disputed. The site where the city of Baghdad developed has been populated for millennia. By the 8th century AD, several villages had developed there, including a Persian hamlet called Baghdad, the name which would come to be used for the Abbasid metropolis. Arab authors, realizing the pre-Islamic origins of Baghdad's name looked for its roots in Persian, they suggested various meanings, the most common of, "bestowed by God". Modern scholars tend to favor this etymology, which views the word as a compound of bagh "god" and dād "given", In Old Persian the first element can be traced to boghu and is related to Slavic bog "god", while the second can be traced to dadāti.
A similar term in Middle Persian is the name Mithradāt, known in English by its Hellenistic form Mithridates, meaning "gift of Mithra". There are a number of other locations in the wider region whose names are compounds of the word bagh, including Baghlan and Bagram in Afghanistan or a village called Bagh-šan in Iran; the name of the town Baghdati in Georgia shares the same etymological origins. A few authors have suggested older origins for the name, in particular the name Bagdadu or Hudadu that existed in Old Babylonian, the Babylonian Talmudic name of a place called "Baghdatha"; some scholars suggested Aramaic derivations. When the Abbasid caliph, al-Mansur, founded a new city for his capital, he chose the name Madinat al-Salaam or City of Peace; this was the official name on coins and other official usage, although the common people continued to use the old name. By the 11th century, "Baghdad" became the exclusive name for the world-renowned metropolis. After the fall of the Umayyads, the first Muslim dynasty, the victorious Abbasid rulers wanted their own capital from which they could rule.
They chose a site north of the Sassanid capital of Ctesiphon, on 30 July 762 the caliph Al-Mansur commissioned the construction of the city. It was built under the supervision of the Barmakids. Mansur believed that Baghdad was the perfect city to be the capital of the Islamic empire under the Abbasids. Mansur loved the site so much he is quoted saying: "This is indeed the city that I am to found, where I am to live, where my descendants will reign afterward"; the city's growth was helped by its excellent location, based on at least two factors: it had control over strategic and trading routes along the Tigris, it had an abundance of water in a dry climate. Water exists on both the north and south ends of the city, allowing all households to have a plentiful supply, uncommon during this time. Baghdad eclipsed Ctesiphon, the capital of the Sassanians, located some 30 km to the southeast. Today, all that remains of Ctesiphon is the shrine town of Salman Pak, just to the south of Greater Baghdad.
Ctesiphon itself had replaced and absorbed Seleucia, the first capital of the Seleucid Empire, which had earlier replaced the city of Babylon. According to the traveler Ibn Battuta, Baghdad was one of the largest cities, not including the damage it has received; the residents are Hanbal. Bagdad is home to the grave of Abu Hanifa where there is a cell and a mosque above it; the Sultan of Bagdad, Abu Said Bahadur Khan, was a Tartar king. In its early years, the city was known as a deliberate reminder of an expression in the Qur'an, when it refers to Paradise, it took four years to build. Mansur assembled engineers and art constructionists from around the world to come together and draw up plans for the city. Over 100,000 construction workers came to survey the plans. July was chosen as the starting time because two astrologers, Naubakht Ahva
Muhammad ibn Musa al-Khwarizmi
Muḥammad ibn Mūsā al-Khwārizmī Latinized as Algorithmi, was a Persian scholar who produced works in mathematics and geography under the patronage of the Caliph Al-Ma'mun of the Abbasid Caliphate. Around 820 AD he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad. Al-Khwarizmi's popularizing treatise on algebra presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications; because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing", he has been described as the father or founder of algebra. The term algebra itself comes from the title of his book, his name gave rise to the terms algorithm. His name is the origin of guarismo and of algarismo, both meaning digit. In the 12th century, Latin translations of his textbook on arithmetic which codified the various Indian numerals, introduced the decimal positional number system to the Western world.
The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical text-book of European universities. In addition to his best-known works, he revised Ptolemy's Geography, listing the longitudes and latitudes of various cities and localities, he further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial. Few details of al-Khwārizmī's life are known with certainty, he was born into a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan. Muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy; the epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul, a viticulture district near Baghdad. However, Rashed suggests: There is no need to be an expert on the period or a philologist to see that al-Tabari's second citation should read "Muhammad ibn Mūsa al-Khwārizmī and al-Majūsi al-Qutrubbulli," and that there are two people between whom the letter wa has been omitted in an early copy.
This would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī even the origins of his knowledge, had not been made. G. J. Toomer... with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmī's religion, Toomer writes: Another epithet given to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was an adherent of the old Zoroastrian religion; this would still have been possible at that time for a man of Iranian origin, but the pious preface to al-Khwārizmī's Algebra shows that he was an orthodox Muslim, so al-Ṭabarī's epithet could mean no more than that his forebears, he in his youth, had been Zoroastrians. Ibn al-Nadīm's Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833. After the Muslim conquest of Persia, Baghdad became the centre of scientific studies and trade, many merchants and scientists from as far as China and India traveled to this city, as did al-Khwārizmī.
He worked in Baghdad as a scholar at the House of Wisdom established by Caliph al-Ma’mūn, where he studied the sciences and mathematics, which included the translation of Greek and Sanskrit scientific manuscripts. Douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmī's contributions to mathematics, geography and cartography established the basis for innovation in algebra and trigonometry, his systematic approach to solving linear and quadratic equations led to algebra, a word derived from the title of his book on the subject, "The Compendious Book on Calculation by Completion and Balancing". On the Calculation with Hindu Numerals written about 820, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe, it was translated into Latin as Algoritmi de numero Indorum. Al-Khwārizmī, rendered as Algoritmi, led to the term "algorithm".
Some of his work was based on Persian and Babylonian astronomy, Indian numbers, Greek mathematics. Al-Khwārizmī corrected Ptolemy's data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea and Africa, he wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a world map for al-Ma'mun, the caliph, overseeing 70 geographers. When, in the 12th century, his works spread to Europe through Latin translations, it had a profound impact on the advance of mathematics in
History of optics
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, followed by theories on light and vision developed by ancient Greek philosophers, the development of geometrical optics in the Greco-Roman world. The word optics is derived from the Greek term τα ὀπτικά meaning "appearance, look". Optics was reformed by the developments in the medieval Islamic world, such as the beginnings of physical and physiological optics, significantly advanced in early modern Europe, where diffractive optics began; these earlier studies on optics are now known as "classical optics". The term "modern optics" refers to areas of optical research that developed in the 20th century, such as wave optics and quantum optics; the earliest known lenses were made from polished crystal quartz, have been dated as early as 750 BC for Assyrian lenses such as the Nimrud / Layard lens. There are many similar lenses from ancient Egypt and Babylon; the ancient Romans and Greeks filled glass spheres with water to make lenses.
However, glass lenses were not thought of until the Middle Ages. Some lenses fixed in ancient Egyptian statues are much older. There is some doubt as to whether or not they qualify as lenses, but they are undoubtedly glass and served at least ornamental purposes; the statues appear to be anatomically correct schematic eyes. In ancient India, the philosophical schools of Samkhya and Vaisheshika, from around the 6th–5th century BC, developed theories on light. According to the Samkhya school, light is one of the five fundamental "subtle" elements out of which emerge the gross elements. In contrast, the Vaisheshika school gives an atomic theory of the physical world on the non-atomic ground of ether and time; the basic atoms are those of earth, water and air, that should not be confused with the ordinary meaning of these terms. These atoms are taken to form binary molecules. Motion is defined in terms of the movement of the physical atoms. Light rays are taken to be a stream of high velocity of tejas atoms.
The particles of light can exhibit different characteristics depending on the speed and the arrangements of the tejas atoms. Around the first century BC, the Vishnu Purana refers to sunlight as "the seven rays of the sun". In the fifth century BC, Empedocles postulated, he believed that Aphrodite made the human eye out of the four elements and that she lit the fire in the eye which shone out from the eye making sight possible. If this were true one could see during the night just as well as during the day, so Empedocles postulated an interaction between rays from the eyes and rays from a source such as the sun, he stated. In his Optics Greek mathematician Euclid observed that "things seen under a greater angle appear greater, those under a lesser angle less, while those under equal angles appear equal". In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles.
Pappus believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis of Ptolemy. In 55 BC, Lucretius, a Roman who carried on the ideas of earlier Greek atomists, wrote: The light and heat of the sun. Despite being similar to particle theories of light, Lucretius's views were not accepted and light was still theorized as emanating from the eye. In his Catoptrica, Hero of Alexandria showed by a geometrical method that the actual path taken by a ray of light reflected from a plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation. In the second century Claudius Ptolemy, in his Optics undertook studies of reflection and refraction, he measured the angles of refraction between air and glass, his published results indicate that he adjusted his measurements to fit his assumption that the angle of refraction is proportional to the angle of incidence.
The Indian Buddhists, such as Dignāga in the 5th century and Dharmakirti in the 7th century, developed a type of atomism, a philosophy about reality being composed of atomic entities that are momentary flashes of light or energy. They viewed light as being an atomic entity equivalent to energy, similar to the modern concept of photons, though they viewed all matter as being composed of these light/energy particles; the early writers discussed here treated vision more as a geometrical than as a physical, physiological, or psychological problem. The first known author of a treatise on geometrical optics was the geometer Euclid. Euclid began his study of optics as he began his study of geometry, with a set of self-evident axioms. Lines can be drawn in a straight line to the object; those lines falling upon an object form a cone. Those things upon which the lines fall are seen; those things seen under a larger angle appear larger. Those things seen by a higher ray, appear higher. Right and left rays appear left.
Things seen within several angles appear clearer. Euclid did not define the physical nature of these visual rays but, using the principles of geometry, he discussed the effects of perspective and the rounding of things seen at a distance. Where Eucl
The Persians are an Iranian ethnic group that make up over half the population of Iran. They share a common cultural system and are native speakers of the Persian language, as well as related languages; the ancient Persians were a nomadic branch of the ancient Iranian population that entered the territory of modern-day Iran by the early 10th century BC. Together with their compatriot allies, they established and ruled some of the world's most powerful empires, well-recognized for their massive cultural and social influence covering much of the territory and population of the ancient world. Throughout history, the Persians have contributed to various forms of art and science, own one of the world's most prominent literatures. In contemporary terminology, people of Persian heritage native to present-day Afghanistan and Uzbekistan are referred to as Tajiks, whereas those in the eastern Caucasus, albeit assimilated, are referred to as Tats; however the terms Tajik and Persian were synonymous and were used interchangeably, many of the most influential Persian figures hailed from outside Iran's present-day borders to the northeast in Central Asia and Afghanistan and to a lesser extent to the northwest in the Caucasus proper.
In historical contexts in English, "Persians" may be defined more loosely to cover all subjects of the ancient Persian polities, regardless of ethnic background. The English term Persian derives from Latin Persia, itself deriving from Greek Persís, a Hellenized form of Old Persian Pārsa. In the Bible, it is given as Parás —sometimes Paras uMadai —within the books of Esther, Daniel and Nehemya. A Greek folk etymology connected the name to a legendary character in Greek mythology. Herodotus recounts this story, devising a foreign son, from whom the Persians took the name; the Persians themselves knew the story, as Xerxes I tried to use it to suborn the Argives during his invasion of Greece, but failed to do so. Although Persis was one of the provinces of ancient Iran, varieties of this term were adopted through Greek sources and used as an official name for all of Iran for many years. Thus, in the Western world, the term Persian came to refer to all inhabitants of the country; some medieval and early modern Islamic sources used cognates of the term Persian to refer to various Iranian peoples, including the speakers of the Khwarezmian language, the Mazanderani language, the Old Azeri language.
10th-century Iraqi historian Al-Masudi refers to Pahlavi and Azari as dialects of the Persian language. In 1333, medieval Moroccan traveler and scholar Ibn Battuta referred to the people of Kabul as a specific sub-tribe of Persians. Lady Mary Sheil, in her observation of Iran during the Qajar era, describes Persians and Leks to identify themselves as "descendants of the ancient Persians". On March 21, 1935, the former king of Iran, Reza Shah of the Pahlavi dynasty, issued a decree asking the international community to use the term Iran, the native name of the country, in formal correspondence. However, the term Persian is still used to designate the predominant population of the Iranian peoples living in the Iranian cultural continent; the earliest known written record attributed to the Persians is from the Black Obelisk of Shalmaneser III, an Assyrian inscription from the mid-9th century BC, found at Nimrud. The inscription mentions Parsua as a tribal chiefdom in modern-day western Iran; the ancient Persians were a nomadic branch of the Iranian population that, in the early 10th century BC, settled to the northwest of modern-day Iran.
They were dominated by the Assyrians for much of the first three centuries after arriving in the region. However, they played a major role in the downfall of the Neo-Assyrian Empire; the Medes, another branch of this population, founded the unified empire of Media as the region's dominant cultural and political power in c. 625 BC. Meanwhile, the Persian dynasty of the Achaemenids formed a vassal state to the central Median power. In c. 552 BC, the Achaemenids began a revolution which led to the conquest of the empire by Cyrus II in c. 550 BC. They spread their influence to the rest of what is called the Iranian Plateau, assimilated with the non-Iranian indigenous groups of the region, including the Elamites and the Mannaeans. At its greatest extent, the Achaemenid Empire stretched from parts of Eastern Europe in the west, to the Indus Valley in the east, making it the largest empire the world had yet seen; the Achaemenids developed the infrastructure to support their growing influence, including the creation of Pasargadae and the opulent city of Persepolis.
The empire extended as far as the limits of the Greek city states in modern-day mainland Greece, where the Persians and Athenians influenced each other in what is a reciprocal cultural exchange. Its legacy and impact on the kingdom of Macedon was notably huge for centuries after the withdrawal of the Persians from Europe following the Greco-Persian Wars; the empire collapsed in 330 BC following the conquests of Alexander the Great, but reemerged shortly after as the Parthian Empire. During the Achaemenid era, Persian colonists settled in Asia Minor. In Lydia, near Sardis, there was the Hyrcanian plain, according to Strabo, got its name from the Persian settlers that were moved from Hyrcania. Near Sardis, there was the plain of Cyrus, which further signified the presence of numerous Persian settlements in
Mathematics in medieval Islam
Mathematics during the Golden Age of Islam during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra, advances in geometry and trigonometry. Arabic works played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries; the study of algebra, the name of, derived from the Arabic word meaning completion or "reunion of broken parts", flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a scholar in the House of Wisdom in Baghdad, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi deals with ways to solve for the positive roots of first and second degree polynomial equations, he introduces the method of reduction, unlike Diophantus, gives general solutions for the equations he deals with.
Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, syncopated, meaning that some symbolism is used; the transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn al-Banna' al-Marrakushi and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī. On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F. Robertson said: "Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra, it is important to understand just. It was a revolutionary move away from the Greek concept of mathematics, geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc. to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, provided a vehicle for the future development of the subject.
Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before." Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. Abu Kamil Shuja' wrote a book of algebra accompanied with geometrical proofs, he enumerated all the possible solutions to some of his problems. Abu al-Jud, Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation. Omar Khayyam found the general geometric solution of a cubic equation. Omar Khayyam wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of cubic or third-order equations, going beyond the Algebra of al-Khwārizmī. Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections; this method had been used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al-Dīn al-Ṭūsī developed a novel approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value.
For example, to solve the equation x 3 + a = b x, with a and b positive, he would note that the maximum point of the curve y = b x − x 3 occurs at x = b 3, that the equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a. His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them; the earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite. The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. In between, implicit proof by induction for arithmetic sequences was introduced by al-Karaji and continued by al-Samaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle; the Greeks had discovered irrational numbers, but were not happy with them and only able to cope by drawing a distinction between magnitude and number.
In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically. Islamic mathematicians including Abū Kāmil Shujāʿ ibn Aslam and Ibn Tahir al-Baghdadi removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations, they worked with irrationals as mathematical objects, but they did not examine their nature. In the twelfth century, Latin translations of Al-Khwarizmi's Arithmetic on the Indian numerals introduced the decimal positional number system to the Western world, his Compendious Book on Calculation by Completion and Balancing presented the first systematic s