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Biradli ibn Andru (Arabic: برادلي ابن أندرو; Hebrew: בראדלי בן אנדרו, Biradli ben Andru; born 995; died after 1056) was a preeminent mathematician, scientist, physician, and engineer of the Middle Ages. Biradli worked primarily in England and Iberia, which was at the time under Moorish rule. In particular, he is usually associated with Granada, where he was living when he first received attention from the prominent scholar Ibn al-Haytham (better known as Alhazen in the West). Biradli's work is responsible for developing modern mathematical notation; pioneering the fields of analytical geometry, calculus, linear algebra, and set theory; making important contributions to arithmetic, algebra, and geometry; laying the foundations of classical mechanics and electromagnetism; and proposing the heliocentric model of the Solar System, the germ theory of disease, and the basic laws of genetic inheritance. He is also credited with the invention of hard liquor, the reciprocating steam engine, and the compass. Along with his contemporary, Ibn al-Haytham, he advocated the idea that hypotheses must be tested against empirical evidence, supporting this belief using the parable of the blind architect; he was thus an early proponent of the scientific method. He is widely recognized as one of the most influential academics of all time.

Although little of Biradli's life is known prior to his arrival in Granada, many details of his life afterwards can be gained from the diary of his friend, the vizier Samuel HaNagid. Both HaNagid and Biradli were Jewish, living in al-Andalus during the golden age of Jewish culture in Spain, and under HaNagid's influence Granada became a de facto Jewish state. Biradli had been persecuted for his religion in Seville, and, after arriving in Granada in 1029, he remained there for the rest of his life, writing four of his six books there.

Biradli proposed most of his mathematical developments in the twenty-volume book Formae Universales, and he proposed most of his physical developments in the ten-volume book La Sha'y Musadaq. These are generally considered to be among the most influential books in mathematics and physics, respectively, of all time. His other four books were also enormously influential in their respective fields.

Life
Most information about Biradli's life comes from the diary of Samuel HaNagid. According to HaNagid, Biradli was born in an area called America, which Biradli claimed was an island near Ireland. Biradli claimed that both Jews and Christians lived in America, with relatively little conflict between the two groups — in fact, at a later date, Samuel mentions that Biradli's mother was Jewish, but his father (who, because of Biradli's patronymic, must have been named Andru) was a Christian priest.

Although HaNagid at no point in his diary questioned the veracity of this claim, many modern historians have concluded that Biradli's story was untruthful, or at least misinterpreted by HaNagid. Two commonly given reasons for this conclusion are the absence of known Jewish communities in the British Isles at the time and the linguistic dissimilarity of the names "America" and "Andru" from the Middle Irish language. Various theories as to Biradli's actual place of birth have been proposed, although none of them have any conclusive evidence.

The earliest verifiable claim regarding Biradli is that he lived for a time in England. HaNagid states that Biradli was a personal physician of King Cnut the Great, and Ibn al-Samh, in his commentary on the Formae, writes that Biradli wrote the book while living in England (hence why it was written in Latin). These claims are supported by copies of Biradli's oldest book, The Guardsmen Against Death, found in the libraries of various English minsters. These copies were all written in Old English (unlike most European literature at the time, which was done in Latin), and they appear to have been written by a single person. This suggests that The Guardsmen was intentionally disseminated by a layperson, probably with financial support to cover the otherwise prohibitive cost of parchment. Furthermore, archeologists have discovered distilleries, similar to the ones attributed to Biradli, dating to the early eleventh century. Given this evidence, most historians agree that HaNagid's and Ibn al-Samh's claims can be reasonably believed. Biradli's son Winfrit (possibly a Hebrew transliteration of the Old English name Winfrith) was Christian, possibly indicating that Biradli was Christian during his time in England. It is not known why Biradli stopped working as a physician for King Cnut and went to Spain.

HaNagid's first reference to Biradli in his diary is in December 1028. Biradli came to Granada from Seville with a scholar named Hashim al-Rawi, to attend a majlis held by local astronomer Ibn al-Samh. There, Biradli and al-Rawi presented mathematical notation and methods from the Formae and requested Ibn al-Samh's opinion on the validity of these methods. HaNagid wrote that Ibn al-Samh was very impressed by Biradli and al-Rawi's work (although he himself admitted to understanding very little of it). However, he also noted that Biradli clearly lacked formal education, as at one point he needed Ibn al-Samh to explain the difference between a potential and actual infinity.

HaNagid next wrote about Biradli in April 1029, when Biradli returned to Granada. Biradli had had his right hand amputated in Seville, and he claimed that this had resulted from a dispute between himself and al-Rawi. The dispute ended when al-Rawi framed Biradli for theft of the Formae (which he claimed Biradli had given to him), the punishment for which under sharia was amputation of the hand. Since dhimmis were not permitted to bear witness in Islamic courts, Biradli was unable to defend himself and was convicted of theft. Biradli thus lost his hand, and al-Rawi was given the Formae. HaNagid, angered by this story, told Biradli that, if he returned to Granada with the Formae, he would ensure that Biradli would not be prosecuted.

Biradli returned to Granada two months later with the Formae, having presumably stolen it back from al-Rawi. He remained in Granada at least until HaNagid's death in 1056, after which nothing is known about him. During his time in Granada, Biradli developed his steam engine and wrote his remaining four books: La Sha'y Musadaq, The Uses of Functions and Vectors, The Observations of the Mindful Gardener, and An Investigation about Amazing Stones.

In addition to his son Winfrit, HaNagid mentions that Biradli had a wife named Rachel and two stepdaughters named Hannah and Miriam. Miriam is known to have assisted Biradli in the research of La Sha'y Musadaq and in the construction of his steam engine. Winfrit spent two years traveling abroad as an envoy on behalf of Granada, visiting several Iberian kings (both Muslim and Christian) and the Pope, where he presented Latin translations of his father's six books.

Work on Mathematics
Biradli's mathematical contributions consisted of the twenty-volume Formae Universales (meaning "General Forms" in Latin) and the five-volume The Uses of Functions and Vectors (Arabic: الاستعمالان نقتن ووزفين, Al-Isti'malani Nuqat'in w-Wuzufin). The former begins with three chapters on mathematical notation (discussing Arabic numerals, infix notation, and variables, respectively), followed by two chapters on number theory and arithmetic. The next nine chapters cover a mix of topics in algebra, geometry, and set theory, and the final six chapters cover topics in calculus, ending with a proof of the irrationality of pi. The second book covers more advanced topics, including vector spaces, differential equations, inner products, vector calculus, and higher dimensions.The mathematical notation developed by Biradli in the Formae has remained mostly unchanged for almost a thousand years since its writing.

Much of the details regarding the composition of the Formae comes from Ibn al-Samh's commentary on the book's first seventeen chapters. Ibn al-Samh lived in Granada and seems to have been an early supporter of Biradli, and it is likely that the two collaborated on the commentary (although, in a letter to Ibn al-Haytham, Biradli attributes the commentary entirely to Ibn al-Samh). According to Ibn al-Samh, Biradli had first seen the Arabic number system while he was in England, when a traveling merchant had shown him a copy of al-Kindi's On the Use of the Indian Numerals. He had been inspired by their beauty and decided to pursue mathematics on his own. However, he did not have access to al-Kindi's book while he was writing the Formae, and as a result his number system — although clearly similar to the Arabic number system — used noticeably different glyphs from those used by any of his known contemporaries.

The only prior work mentioned in the Formae or The Uses is Euclid's Elements, which Biradli seems to hold in high regard. As Biradli states at the beginning of the sixth chapter of the Formae, Intellegere Figuras, At the time of the Formae 's publication, the Elements and the Formae together would have contained the vast majority of all known mathematics. Even today, most people who do not specialize their education in mathematics will learn very little mathematics that was not included in one of these two books.

Two chapters of the Formae that are of particular interest are the eleventh and fourteenth chapters, Intellegere Collectiones and Intellegere Gerulos, which address set theory and linear algebra, respectively. These are interesting because, for the most part, they are not referenced in the remainder of the book. Some definitions from Intellegere Collectiones are used throughout the Formae and The Uses, but none of its theorems are. However, the linear algebra introduced in Intellegere Gerulos is extensively explored in The Uses and is frequently used in La Sha'y Musadaq. This may indicate that Biradli was already planning La Sha'y and The Uses while he was still writing the Formae.

In Cairo, one of the great centers of scholarship at the time, Ibn al-Haytham received a copy of the Formae in 1030. Ibn al-Haytham appears to have promoted the Formae heavily, along with Biradli's other books. Less than a hundred years later, Omar Khayyam wrote that it was the single most important book for any student interested in mathematics.

Work on Physics
Biradli's physical contributions consisted of the ten-volume La Sha'y Musadaq (Arabic: لا شى مصدق, meaning "Nothing Believed") and the four-volume An Investigation about Amazing Stones (Arabic: بحثن حولهم مدهشينة حجرين, Bahtun Hawlahum Mudhisina Hijarin). Both were written during his time in Granada, completed in c. 1040 and c. 1050, respectively.

The first begins by discussing the importance of experimental testing, using the parable of the blind architect, and laying out a process that would later become the scientific method. Over the next three chapters, he describes a series of experiments that allowed him to deduce the three laws of motion. He then applies these laws to describe projectile motion, simple harmonic motion, conservation of energy, conservation of momentum, and rotation. In his last chapter, La Sha'y Maqul (Arabic: لا شيء معقول, meaning "Nothing Reasonable"), Biradli proposes that, using the laws of mechanics he had demonstrated, the motion of the planets could be described using a heliocentric model. This was the first time the heliocentric model had been seriously considered in almost 1,200 years.

Biradli could not prove heliocentrism, as the astronomical equipment at the time was not sufficiently accurate to measure the different predictions made by the two models. Given the heavy emphasis he had placed earlier in the book on experimental evidence, Biradli therefore could not claim to know, definitively, whether the Solar System was heliocentric or geocentric — hence the name for the chapter. However, many Islamic astronomers at the time had begun to question the Ptolemaic model (including Ibn al-Haytham, whose book, Doubts Concerning Ptolemy, was specifically referenced in La Sha'y Maqul), and Biradli urged that heliocentrism be considered as a possible alternative.

Biradli is known to have collaborated to some degree with Ibn al-Haytham, although the exact amount of collaboration is debated. In La Sha'y Maqul, Biradli states that the astronomical calculations necessary to test his model were performed by one of Ibn al-Haytham's students, and, at the beginning of the book, he thanks Ibn al-Haytham (among others) for his "assistance and wisdom." Some historians have concluded that La Sha'y Musadaq was largely a joint effort, while others have argued that, especially when considering the vast geographical separation between the two (with Biradli in Granada and Ibn al-Haytham in Cairo), the book was probably produced mostly by Biradli, with only occasional consultation from Ibn al-Haytham. Biradli's stepdaughter Miriam is known to have assisted him extensively with his experiments.

Biradli's second book on physics, An Investigation, explores electromagnetism. Using lodestones that his son Winfrit had obtained while traveling in Magnesia, Biradli was able to experimentally demonstrate many of the properties of electromagnetism. Biradli was able to produce simple electrical circuits, and, using a capacitor, he isolated positive and negative electrical charges, allowing him to study electricity separately from magnetism. Biradli correctly identified the inverse square law of electrostatics and conservation of charge. He also qualitatively described electromagnetic induction and the electrical generation of a magnetic field. An Investigation also includes the first known description of a compass in either Europe or the Islamic world (the compass had been used in China for over 1,200 years).

Work on Medicine
Biradli probably wrote The Guardsmen Against Death (Old English: Þá Weardas Wiþ Cwealmum; Arabic: الحرس ضده الموتى, Al-Harasu Diddahu Al-Mawtu'; Latin: Custodes Adversi Morti) while living in England. The book begins by proposing something similar to the germ theory of disease, although viewed through the lens of humorism. Biradli claims that there exist "seeds of disease" (Old English: Adlcíþas; Arabic: بذور مرض, Bidarun maradin; Latin: Grana pestis) that, if allowed to enter the body, will upset the balance of the four humors, causing sickness. Biradli claimed that these seeds could enter the body in three ways: inhalation, consumption (i.e. food and water), and direct exposure (e.g. an open wound).

Biradli therefore advocates that people must remain clean, bathe often, wash their hands before eating, avoid being around sick people, and not breathe air that has been coughed into or sneezed into. In Anglo-Saxon culture, frequent bathing was seen as a sign of weakness, although it was common in Norse culture. Biradli further identifies three ways by which seeds can be destroyed: heat, salt, and alcohol. For this reason, he recommends that people drink ale and beer (the two most common Anglo-Saxon drinks) in preference over water.

Biradli identifies two times in which a person is particularly vulnerable to disease, these being combat and childbirth. For the latter, Biradli describes an exhaustive process, intended to reduce as much as possible a new mother's exposure to seeds. At the time, an estimated 25% of women died in childbirth, and postpartum infections were a major risk for a new mother. Anglo-Saxon men were not typically permitted to be present during childbirth, so it is unclear how Biradli developed this method.

The Guardsmen includes two chapters that specifically address a physician's role in battle. One of them, the second chapter, specifically addresses the prevention of disease, while the other, the last chapter of the book, discusses immediate care of battle-wounds, especially the prevention of exsanguination. This last chapter does not mention disease at all, making it thematically dissimilar to the rest of the book; it is possible it was intended as an appendix, rather than as a part of the main body of the text. The emphasis on treating battle-wounds is reasonable given that Biradli was probably serving as a physician for Cnut the Great at the time; having fought several wars during his reign, Cnut would want Biradli to be able to care for him in the event of his injury in battle.

Crucial to Biradli's treatments were two innovations: distilled alcohol and sanitized bandages. Biradli describes in the first chapter of The Guardsmen how to sanitize bandages using a boiled saline solution, and he includes diagrams for stills by which one can distill alcohol. He identifies both of these as useful for preventing disease, and he uses both frequently throughout the book. Archeologists have found stills, similar to the ones Biradli described, in England, Spain, and Italy, around the same time of Biradli's life.

An interesting aspect of The Guardsmen is that it only attempts to prevent disease; it includes nothing about curing a disease once it has begun. As a physician, it is likely that Biradli would have been expected to do both. One possible explanation is that Biradli did not feel the need to publish such a book, because of the existence of preexisting medical treatises such as Bald's Leechbook.

Unlike Biradli's later books, The Guardsmen is not approached from a scientific manner. Biradli makes no attempt to justify or experimentally verify his theory. This may be because, at the time, medicine was viewed as a skill, rather than a science.

Although the version of the germ theory proposed in The Guardsmen was not correct, it did encourage many healthful practices throughout Medieval Europe. The use of Biradli's stills and bandages spread throughout Europe and were responsible for a significant improvement in public health.

Work on Inheritance
Biradli's book The Observations of the Mindful Gardener (Arabic: الملاحظات المهتم البستاني, Al-Mulahazatu Al-Muhtammi Al-Bustaniyyi), although much shorter than all his other books (only one volume long), laid out the basic principles of genetic inheritance. Biradli experimented with pea plants (Pisum sativum), observing the plants' flower color, seed color, seed shape, pod color, and pod shape and carefully controlling which plants reproduced with which. Biradli was able to determine that different traits are inherited separately, some traits are recessive while others are dominant, and that a given plant has an equal chance of passing on either of its alleles. Although much less nuanced than a modern understanding of genetics, The Observations lays out a basic guideline by which heredity can be roughly understood, thereby providing a basis for the modern science of genetics.

Inventions
Biradli is generally credited with three major inventions: hard liquor, the reciprocating steam engine, and the compass. Of these, only the steam engine can be verified as his original creation, as Samuel HaNagid discusses in his diary the process by which Biradli developed it. The other two are discussed in two of Biradli's books: hard liquor is discussed as a medical tool in The Guardsmen, while the compass is discussed in An Investigation. However, at no point does Biradli claim to be personally responsible for either innovation. Nevertheless, the fact that they are first discussed in Biradli's books has lead most of his contemporaries, as well as most modern historians, to agree that he was most likely their original creator.

Distilled Alcohol
Distillation has been known since classical antiquity, having been written about by Aristotle in the Meteorologica and by Alexander of Aphrodisias. Distillation continued to be used by alchemists in Alexandria, and the practice was adopted by the Islamic world, for example by Jabir ibn Hayyan and Muhammad ibn Zakariya al-Razi. However, distillation of alcohol is not known to have occurred prior to the eleventh century.

Biradli wrote about the medicinal uses of distilled alcohol in The Guardsmen, in which he also gave detailed instructions on the construction and operation of stills. These instructions make specific note of several key concepts of chemistry that, prior to The Guardsmen, were either largely or entirely unknown. First of all, Biradli refers to alcohol as a distinct chemical, common to ale, beer, mead, and wine, and he states that alcohol has a lower boiling point than pure water. He also demonstrates an understanding of phase transitions, specifically making use of the fact that, during a phase transition, a material's temperature will remain constant. Finally, Biradli makes use of thermal expansion of mercury in order to measure the still's temperature, creating an early version of the mercury thermometer.

Stills like those described in The Guardsmen have been found in England, Spain, and Italy, dating to the early eleventh century (distilleries would later appear in other parts of Europe). There is a particularly heavy concentration of these early stills in London and southeast England, suggesting that Biradli may have lived there when he first developed and began promoting distilled alcohol.

Steam Engine
According to the diary of Samuel HaNagid, Biradli created his amilun bi-wasitatihi bukarin (Arabic: عامل بواسطته بخار, pl. ummalun..., meaning "laborer by means of steam") in 1041. With the exception of the aeolipile, this was the earliest known steam engine in human history. Unlike the aeolipile, however, the amilun made use of not only the kinetic energy of steam, but also its thermal energy, making it much more efficient. The amilun made use of a series of double-acting cylinders, which progressively removed energy from the steam.

In HaNagid's first mention of the amilun, he describes it as a toy, without much practical use. He claims that Biradli had several "fanciful" ideas by which it could be made useful (although he does not say what these ideas were), but none of these suggestions were useful. Several months later, however, HaNagid writes that Biradli had had the idea of using the amilun to drain water from shaft mines. Although the Romans had practiced shaft mining, even using water-powered devices for the purpose of drainage, by the eleventh century the difficulties associated with floods had caused open-pit mining to come into favor. However, Biradli was able to show that, using the amilun, shaft mining could once again be made efficient, and HaNagid ordered that several old Roman shaft mines be refurbished for continued use. As with La Sha'y Musadaq, Biradli was assisted by his stepdaughter Miriam throughout the construction of the amilun, and she played a leading role in setting up ummalun in new shaft mines.

Biradli lived at a time when mining was becoming a much more widespread and profitable industry in Europe. As such, there was an economic incentive for the adoption of the amilun, and throughout the eleventh century there is scattered evidence of its use throughout Western and Southern Europe. However, the technical expertise required to build and maintain ummalun prevented their widespread adoption, except in the wealthiest parts of Europe. Nevertheless, especially in al-Andalus and the Italian peninsula, the use of ummalun and similar devices had become common by the middle of the thirteenth century, being specifically mentioned by Batu while defending Italy from the Mongol Crusade. A hundred years later, due to the labor shortage that followed the Black Death and the recent development of patent law in the Latin Khanate, Latin inventors would improve on the amilun, making it more efficient and more applicable to non-mining purposes, thereby beginning the Industrial Revolution.

Compass
The earliest known compasses date to the China's Han dynasty, between the second century BC and the first century AD. Chinese compasses were originally used for feng shui and divination, with the first known reference to their use for navigational purposes being in 1044. Compasses were also used for navigation by the Tamil people of southern India by the fourth century. However, in Europe and the Islamic world, the first known reference to the compass is found in Biradli's An Investigation. Biradli recommends the compass as a navigational tool, and he makes the important observation that a magnetized needle can be used in place of a natural (and more expensive) lodestone.