title: Consumer behaviour

text: The consumer's affective state has implications for a number of different dimensions of consumer behaviour, including information search, evaluation of alternatives, product choice, service encounters, complaining, and advertising responses. Westbrook (1987, p. 259) defines affect as a “class of mental phenomena uniquely characterised by a consciously experienced, subjective feeling state, commonly accompanying emotions and moods." Research suggests that affect plays an important role in underlying attitudes, as well as shaping evaluation and decision-making.


title: Consumer behaviour

text: Consumer researchers have noted the difficulties separating the concepts of affect, emotions, feelings, and mood. The line between emotions and mood is difficult to draw and consumer researchers often use the concepts interchangeably. Yet other researchers note that a detailed understanding of the relationship between affect and consumer behaviour has been hampered by the lack of research in the area. Indeed, within the consumer behaviour literature, there is widespread agreement that the role of emotions is an area that is currently under-researched and is in need of greater attention, both theoretically and empirically.


title: Consumer behaviour

text: Studies have found that people in a positive mood are more efficient at information search activities. That, is they are more efficient at processing information, are able to integrate information by identifying useful relationships and arrive at creative solutions to problems. Due to their efficiency processing information, those who are in a positive mood are generally quicker to make decisions and easier to please. Research consistently shows that people in a positive mood are more likely to evaluate information positively. As online environments become more important as a consumer search tool, it may be prudent for web designers to consider site-design issues such as ease of navigation, lest poor design contribute to customer frustration thereby engendering a bad mood and ultimately leading to unfavourable product/brand evaluations.


title: Consumer behaviour

text: Affect may play an important role in impulse-buying decisions. Research suggests that consumers place higher weightings on immediate affective rewards and punishments, while delayed rewards receive less weighting. For instance, the immediate pleasure of eating a sweet treat often outweighs the longer term benefits of eating a healthy alternative such as fruit. This occurs because the immediate emotional gain is a strong driver, and one that consumers can readily visualise whereas the more distant goal lacks sufficient strength to drive choice.


title: Consumer behaviour

text: Customers who are in a bad mood are more difficult to please. They are slower to process information and consequently take longer to make decisions. They tend to be more argumentative and are more likely to complain.


title: Consumer behaviour

text: The relationship between affect and customer satisfaction is an area that has received considerable academic attention, especially in the services marketing literature. The proposition that there is a positive relationship between affect and satisfaction is well supported in the literature. In a meta-analysis of the empirical evidence, carried out in 2001, Szymanski et al., suggest that affect may be both an antecedent to and an outcome of satisfaction. Emotions elicited during consumption are proposed to leave affective traces in memory that are available for consumers to access and integrate into their satisfaction assessments.


title: Consumer behaviour

text: A 2011 meta-analysis illustrates how both repurchase intent and loyalty enjoy a strong positive relationship (0.54) with customer satisfaction. Another meta-analysis finds that "The results indicate that both cognitive-related variables (including brand awareness, brand personality, and brand identity) and hedonic-related variables (including hedonic attitude, entertainment, and aesthetic appeal) have significant impacts on quality and value perceptions towards the brand (including perceived quality, reputation, brand image, perceived value, commitment, and trust). In addition, these variables are all significant predictors of brand loyalty."


title: Consumer behaviour

text: A third meta-analysis, from 2013 elaborates on the concept of brand personality (bp): "First, the key drivers of BP are communication with hedonic benefit claims, branding activities, a brand’s country-of-origin, and consumer personalities. Second, the study finds that the effects of BP are stronger for mature brands than for brands in the early life cycle stages. Third, sincerity and competence have the strongest influence on brand success variables (e.g., brand attitude, image, commitment, purchase intention), while excitement and ruggedness have the weakest influence on brand attitude and brand commitment."


title: Consumer behaviour

text: Emotion can play an important role in advertising. In advertising, two different approaches to persuasion are common: (a) "thinking ads" that require cognitive processing (also known as the "central route to persuasion") and, (b) "feeling ads" that are processed at an emotional level (also known as the "peripheral route"). Advertisers can bypass cognitive, rational processing which can lead to counter-arguing by simply appealing to the emotions. Neuro-imaging studies suggest that when evaluating brands, consumers primarily use emotions (personal feelings and experiences) rather than information (brand attributes, features, and facts).


title: Consumer behaviour

text: It is relatively widely accepted that emotional responses require fewer processing resources (i.e. are easier) and also result in more enduring associations with the brand being advertised. Feelings elicited by the advertising message can shape attitudes towards the brand and to the advertisement.


title: Consumer behaviour

text: Customer loyalty, defined as “the relationship between an individual's relative attitude and repeat patronage" (Dick and Basu, 1994: p. 99). Thus, by definition, loyalty has both an attitudinal component and a behavioural component. Dick and Basu proposed four types of loyalty based on relative attitude and patronage behaviour:


title: Consumer behaviour

text: Loyalty marketing programs are built on the insight that it costs 5-20 times more to acquire a new customer than to retain an existing customer. Marketers use a variety of loyalty programs to strengthen customer attitudes towards the brand (or service provider/retailer) in order to retain customers, minimise customer defections, and strengthen loyalty bonds with existing customers. Broadly there are two types of program: reward and recognition programs. In a Reward Program, the customer accumulates points for each purchase, and the points can subsequently be exchanged for goods or services. Recognition Programs operate on a quasi-membership basis where the consumer is issued with a card that upon presentation leads to various entitlements such as free upgrades, special privileges, or access to products/services that are not normally available to non- members, and that acknowledge the loyal customer's "VIP" status. For example, a hotel might recognise loyal patrons by providing a complimentary fruit bowl and bottle of champagne in the room on arrival. Whereas reward programs are motivated by the consumer's desire for material possessions, recognition programs are motivated by the consumer's need for esteem, recognition, and status. Many commercial loyalty programs are hybrid schemes, combining elements of both reward and recognition. In addition, not all reward programs are designed to encourage loyalty. Certain reward programs are designed to encourage other types of positive customer behaviour such as the provision of referrals or providing positive word-of-mouth (WOM) recommendations.


title: Consumer behaviour

text: Loyalty marketing can involve the use of databases and sophisticated software to analyse and profile customer loyalty segments with a view to identifying the most desirable segments, setting goals for each segment, and ultimately attempting to increase the size of the loyal customer base.


title: Consumer behaviour

text: Customer citizenship behaviour refers to actions that are not part of the customer's normal behaviour, that are of a voluntary or discretionary in nature and which are thoughtful, considerate, and helpful. Citizenship behaviour often requires some type of sacrifice on the part of customers. Service marketers are particularly interested in citizenship behaviour because it harnesses the consumer's labour power and therefore increases organisational efficiency. It also has the potential to improve service quality.


title: Consumer behaviour

text: Traditional models of consumer behaviour were developed by scholars such as Fishbein and Ajzen and Howard and Sheth in the 1960s and 70s. More recently, Shun and Yunjie have argued that online consumer behaviour is different to offline behaviour and as a consequence requires new theories or models. After Covid-19, online consumer behavior seems more essential, because since Covid-19 began, there were about 31% more people started shopping online with 43% of all respondents compared to only 12% of respondents before Covid-19.


title: Consumer behaviour

text: Research has identified two types of consumer value in purchasing, namely product value and shopping value. Product value is likely to be similar for both online and offline shoppers. However, the shopping experience will be substantially different for online shoppers. In an offline shopping environment, consumers derive satisfaction from being within the physical store environment or retail landscape (hedonic motivations). In the case of online purchasing, shoppers derive satisfaction from their ability to navigate a website and the convenience of online searching which allows them to compare prices and 'shop around' with minimal time commitment. Thus the online consumer is motivated by more utilitarian factors.


title: Consumer behaviour

text: Consumers may use online platforms for various stages of the purchase decision. Some consumers use online sources simply to acquire information about planned purchases. Others use online platforms for making the actual purchase. In other situations, consumers may also use online platforms to engage in post-purchase behaviours, such as staying connected with a brand by joining a brand community. Or they may become a brand advocate by posting a product review online, or providing brand referrals via social media. Some e-commerce providers have encountered a challenge in courting consumers who seek information online, but still prefer to turn to bricks and mortar retailers for their purchase. To understand the needs and habits of these and other kinds of online shoppers, online marketers have segmented consumers into different kinds of online behaviour in accordance with their online behavioural characteristics. Lewis and Lewis (1997) identified five market segments based on the way that consumers use the Internet in the purchase decision process:


title: Consumer behaviour

text: Wendy Moe (2003) argues that in the offline environment, consumers who are shopping in stores can be easily classified by experienced sales employees only by watching their shopping behaviours. Such classification may not appear online, but Moe and Fader argued that it is feasible to predict practical buying, surfing, and searching action online by investigating click patterns and repetition of visit within online behaviour. In addition, a report of E-consultancy about "benchmarking of user experience" outlined three kinds of online consuming behaviour as a valuable classification for the research of design of web pages to better serve different kinds of consuming behaviour. The three categories are: "trackers", "hunters", and "explorers".


title: Consumer behaviour

text: As the preceding table shows, the first row indicates the process of a consumer buying a new product, while the second and third row illustrates the positive influences the Internet could have on buying process by creating effective communications with online consumers. For example, suppose a consumer carelessly see an advertisement about laptops on Wechat, a popular Chinese social media developed by Tencent. He begins to feel that his laptop is a bit out of date and want to buy a new one, which the outcome of good advertisement placed on the daily Internet tool. He doesn't know anything about how to buy a new one as business change so fast today, so he search on Google to find out the answer. On the result page, what he finds out is the promotional ads which mainly come from JD.com and Taobao, RingToRed.Com two main Chinese competitors of online retailer at this field. As always, he used to prefer JD.com, which provides comparison in detail on brands, price, place and ways of payment and delivery. After careful selection, he makes his order through payment of Wechat, which was placed inside of JD.com. JD.com has one of the fastest distribution channels within China and it support excellent post-purchase service to maintain its position in the market.


title: Consumer behaviour

text: Consumers decide whether or not they like a product within 90 seconds of viewing it for the first time. Therefore, having an aesthetically pleasing product is essential in the marketplace. Studies in processing fluency and consumer behaviour have revealed that “that people prefer visual displays that are easier to process and understand.” and “When a product matches the user’s associations with it it is perceived as more attractive.” Visually fluent products draw upon consumer's pre-existing associations with their design elements, leading to a sense of familiarity and understanding with the product at hand. Visual cues such as colour, composition, typography, and imagery are associated with the phenomenon of fluency.


title: Consumer behaviour

text: Research in colour psychology has shown 62-90% of consumer product assessment is based on colour alone. Indeed, colours have been shown to be linked to consumer's perceptions on a product's quality, reliability, and value. The colours blue and black are viewed as being more reliable, valuable, and expensive while yellow, orange, and brown are associated with cheapness and low quality. Therefore, a product intended to be perceived as "high quality" with a predominately orange and brown palette would lack visual fluency and would likely fail to elicit a positive response with consumers. However, this can be advantageous if the consumer is already in the market for an item that is known to be inexpensive, in which case the use of yellow, orange, or brown would be appropriate.


title: Consumer behaviour

text: Composition is another visual tool that has the ability to affect information processing and influence in consumer perceptions. Studies have shown that consumers in western countries will associate products that are right aligned or placed on the right side of a display to be higher quality. Humans also have a center bias, which makes products that are centered or symmetrical in composition or display seem intrinsically more pleasing. Products that are centered in composition or have centered elements are perceived as being more attractive, popular, and important than products that are left aligned or right aligned. When an object is centered compositionally, it easier for a viewer to interpret and understand. It becomes more “fluent” and is therefore viewed as being more aesthetically pleasing.


title: Consumer behaviour

text: Evidence has shown that pictorial imagery correlates to higher instances of consumer recall and recognition. Pictorial imagery is also easier to process and gains consumer's attention faster. There is significant evidence that when consumers are presented with multiple choices, they will view objects more positively and more aesthetically pleasing when surrounded by congruent imagery. After repeated exposure, this familiar imagery becomes incorporated into consumer's visual lexicon and they become “fluent” in it. Images with higher levels of visual fluency perceived as being more familiar, likeable, and friendly and are therefore more likely to be chosen by consumers.


title: Consumer behaviour

text: Although studies have shown that of pictorial imagery is easier for consumers to process and understand, the choice of typography remains an indispensable element of product design. Handwritten and scripted fonts are associated with individuality, femininity, and luxury while sans serif fonts embody energy, cleanliness, and modernity. Font size has also been shown to have a direct correlation on the emotional attributes assigned to a product. One study has shown that larger type size and weight is perceived as more intimidating and authoritative. Although imagery reigns supreme in product design, it is important to note that type is processed just as easily as pictorial information when the consumer is already familiar with the product.


title: Consumer behaviour

text: Consumer behavior in certain ways affects how much material is used to produce goods, how much material is recycled or composted, how much ends up as pollution, how much ends up in landfills, where goods are produced, how far they travel, and the carbon footprint of manufacturing, transportation, and disposal. Green marketing targets consumers who take the environmental impact of their purchases into account. One 2017 study found no impact of green marketing on consumer behavior in Bangladesh. The study suggests policies be made that decrease the cost of eco-friendly products. It also encourages the implementation of programs which raise consumer awareness regarding the issue of green consumption.


title: Consumer behaviour

text: There are psychological factors which contribute to a consumer's perception surrounding their personal contributions to climate change inducing actions. One of the more well studied biases is referred to as the "better-than-average", or self-enhancing bias. This bias depicts an individual's tendency to perceive that their actions are superior, especially when compared to peers or demographically similar consumers. It has been found that this cognitive bias is indeed present when considering how consumers perceive their pro-environmental efforts. This may be a result of information about climate change leading to feelings of guilt and concern, which activates an unconscious thought process (denial, the better-than-average effect, and other cognitive reactions) that leads to a reduced perception of the threat of climate change. It is a mental defense mechanism that ultimately leads to a reduction in perceived individual responsibility to take part in green behaviors and one-planet-living.


title: Consumer behaviour

text: To gain insights into consumer behaviour, researchers uses the standard battery of market research methods such as surveys, depth interviews, and focus groups. Increasingly, researchers are turning to newer methodologies and technologies in an effort to seek deeper understandings of why consumers behave in certain ways. These newer methods include "ethnographic research" (also known as participant observation) and "neuroscience" as well as experimental lab designs. In addition, researchers often turn to separate disciplines for insights with potential to inform the study of consumer behaviour. For instance, behavioural economics is adding new insights into certain aspects of consumer behaviour.


title: Consumer behaviour

text: Ethnographic research or ethnography has its origins in anthropology. However, marketers use ethnographic research to study the consumer in terms of cultural trends, lifestyle factors, attitudes and the way that social context influences product selection, consumption, and usage. Ethnographic research, also called "participant observation", attempts to study consumer behaviour in natural settings rather than in artificial environment such as labs. Different types of ethnographic research are used in marketing including;


title: Consumer behaviour

text: Trendspotters such as Faith Popcorn's BrainReserve make extensive use of ethnographic research to spot emergent trends.


title: Consumer behaviour

text: Consumer neuroscience (also known as neuromarketing) refers to the commercial use of neuroscience when applied to the investigation of marketing problems and consumer research. Some researchers have argued that the term "consumer neuroscience" is preferred over neuromarketing or other alternatives.


title: Consumer behaviour

text: Consumer neuroscience employs sophisticated bio-metric sensors, such as electroencephalography (EEG), functional magnetic resonance imaging (fMRI) and eye tracking, to study the ways that consumers respond to specific stimuli such as product displays, brands, packaging information, or other marketing signals. Such tests reveal stimuli that trigger the brain's pleasure centre.


title: Consumer behaviour

text: Consumer neuroscience has become a mainstream component of consumer research methods. International market research company Nielsen Research has recently added neuromarketing to its services by acquiring Innerscope, a company specialising in neuromarketing research thus enabling Nielsen to add neuromarketing research to the suite of services available to clients.


title: Pali

text: Pali () is a Middle Indo-Aryan liturgical language native to the Indian subcontinent. It is widely studied because it is the language of the Buddhist "Pāli Canon" or "Tipiṭaka" as well as the sacred language of "Theravāda" Buddhism. Early in the language's history, it was written in the Brahmi script.


title: Pali

text: The word 'Pali' is used as a name for the language of the Theravada canon. The word seems to have its origins in commentarial traditions, wherein the (in the sense of the line of original text quoted) was distinguished from the commentary or vernacular translation that followed it in the manuscript. K. R. Norman suggests that its emergence was based on a misunderstanding of the compound , with being interpreted as the name of a particular language.


title: Pali

text: The name Pali does not appear in the canonical literature, and in commentary literature is sometimes substituted with , meaning a string or lineage. This name seems to have emerged in Sri Lanka early in the second millennium CE during a resurgence in the use of Pali as a courtly and literary language.


title: Pali

text: As such, the name of the language has caused some debate among scholars of all ages; the spelling of the name also varies, being found with both long "ā" and short "a" , and also with either a retroflex or non-retroflex "l" sound. Both the long ā and retroflex are seen in the ISO 15919/ALA-LC rendering, ; however, to this day there is no single, standard spelling of the term, and all four possible spellings can be found in textbooks. R. C. Childers translates the word as "series" and states that the language "bears the epithet in consequence of the perfection of its grammatical structure".


title: Pali

text: There is persistent confusion as to the relation of to the vernacular spoken in the ancient kingdom of Magadha, which was located around modern-day Bihār. Beginning in the Theravada commentaries, Pali was identified with 'Magahi', the language of the kingdom of Magadha, and this was taken to also be the language that the Buddha used during his life. In the 19th century, the British Orientalist Robert Caesar Childers argued that the true or geographical name of the Pali language was Magadhi Prakrit, and that because "pāḷi" means "line, row, series", the early Buddhists extended the meaning of the term to mean "a series of books", so "pāḷibhāsā" means "language of the texts".


title: Pali

text: However, modern scholarship has regarded Pali as a mix of several Prakrit languages from around the 3rd century BCE, combined and partially Sanskritized. There is no attested dialect of Middle Indo-Aryan with all the features of Pali. In the modern era, it has been possible to compare Pali with inscriptions known to be in Magadhi Prakrit, as well as other texts and grammars of that language. While none of the existing sources specifically document pre-Ashokan Magadhi, the available sources suggest that Pali is not equatable with that language.


title: Pali

text: Modern scholars generally regard Pali to have originated from a western dialect, rather than an eastern one. Pali has some commonalities with both the western Ashokan Edicts at Girnar in Saurashtra, and the Central-Western Prakrit found in the eastern Hathigumpha inscription. These similarities lead scholars to associate Pali with this region of western India. Nonetheless, Pali does retain some eastern features that have been referred to as "Māgadhisms".


title: Pali

text: Pāḷi, as a Middle Indo-Aryan language, is different from Classical Sanskrit more with regard to its dialectal base than the time of its origin. A number of its morphological and lexical features show that it is not a direct continuation of Sanskrit. Instead it descends from one or more dialects that were, despite many similarities, different from .


title: Pali

text: The Theravada commentaries refer to the Pali language as "Magadhan" or the "language of Magadha". This identification first appears in the commentaries, and may have been an attempt by Buddhists to associate themselves more closely with the Maurya Empire.


title: Pali

text: However, only some of the Buddha's teachings were delivered in the historical territory of Magadha kingdom. Scholars consider it likely that he taught in several closely related dialects of Middle Indo-Aryan, which had a high degree of mutual intelligibility.


title: Pali

text: Theravada tradition, as recorded in chronicles like the Mahavamsa, states that the "Tipitaka" was first committed to writing during the first century BCE. This move away from the previous tradition of oral preservation is described as being motivated by threats to the "Sangha" from famine, war, and the growing influence of the rival tradition of the Abhayagiri Vihara. This account is generally accepted by scholars, though there are indications that Pali had already begun to be recorded in writing by this date. By this point in its history, scholars consider it likely that Pali had already undergone some initial assimilation with Sanskrit, such as the conversion of the Middle-Indic "bamhana" to the more familiar Sanskrit "brāhmana" that contemporary brahmans used to identify themselves.


title: Pali

text: In Sri Lanka, Pali is thought to have entered into a period of decline ending around the 4th or 5th century (as Sanskrit rose in prominence, and simultaneously, as Buddhism's adherents became a smaller portion of the subcontinent), but ultimately survived. The work of Buddhaghosa was largely responsible for its reemergence as an important scholarly language in Buddhist thought. The "Visuddhimagga", and the other commentaries that Buddhaghosa compiled, codified and condensed the Sinhala commentarial tradition that had been preserved and expanded in Sri Lanka since the 3rd century BCE.


title: Pali

text: With only a few possible exceptions, the entire corpus of Pali texts known today is believed to derive from the Anuradhapura Maha Viharaya in Sri Lanka. While literary evidence exists of Theravadins in mainland India surviving into the 13th Century, no Pali texts specifically attributable to this tradition have been recovered. Some texts (such as the Milindapanha) may have been composed in India before being transmitted to Sri Lanka, but the surviving versions of the texts are those preserved by the Mahavihara in Ceylon and shared with monasteries in Theravada Southeast Asia.


title: Pali

text: The earliest inscriptions in Pali found in mainland Southeast Asia are from the first millennium CE, some possibly dating to as early as the 4th Century. Inscriptions are found in what are now Burma, Laos, Thailand and Cambodia and may have spread from southern India rather than Sri Lanka. By the 11th Century, a so-called "Pali renaissance" began in the vicinity of Pagan, gradually spreading to the rest of mainland Southeast Asia as royal dynasties sponsored monastic lineages derived from the Mahavihara of Anuradhapura. This era was also characterized by the adoption of Sanskrit conventions and poetic forms (such as "kavya") that had not been features of earlier Pali literature. This process began as early as the 5th Century, but intensified early in the second millennium as Pali texts on poetics and composition modeled on Sanskrit forms began to grow in popularity. One milestone of this period was the publication of the Subodhalankara during the 14th Century, a work attributed to Sangharakkhita Mahāsāmi and modeled on the Sanskrit Kavyadarsa.


title: Pali

text: Despite an expansion of the number and influence of Mahavihara-derived monastics, this resurgence of Pali study resulted in no production of any new surviving literary works in Pali. During this era, correspondences between royal courts in Sri Lanka and mainland Southeast Asia were conducted in Pali, and grammars aimed at speakers of Sinhala, Burmese, and other languages were produced. The emergence of the term 'Pali' as the name of the language of the Theravada canon also occurred during this era.


title: Pali

text: While Pali is generally recognized as an ancient language, no epigraphical or manuscript evidence has survived from the earliest eras. The earliest samples of Pali discovered are inscriptions believed to date from 5th to 8th Century located in mainland Southeast Asia, specifically central Siam and lower Burma. These inscriptions typically consist of short excerpts from the Pali Canon and non-canonical texts, and include several examples of the Ye dhamma hetu verse.


title: Pali

text: Surprisingly, the oldest surviving Pali manuscript was discovered in Nepal dating to the 9th Century. It is in the form of four palm-leaf folios, using a transitional script deriving from the Gupta script to scribe a fragment of the Cullavagga. The oldest known manuscripts from Sri Lanka and Southeast Asia date to the 13th-15th Century, with few surviving examples. Very few manuscripts older than 400 years have survived, and complete manuscripts of the four Nikayas are only available in examples from the 17th Century and later.


title: Pali

text: Pali was first mentioned in Western literature in Simon de la Loubère's descriptions of his travels in the kingdom of Siam. An early grammar and dictionary was published by Methodist missionary Benjamin Clough in 1824, and an initial study published by Eugène Burnouf and Christian Lassen in 1826 ("Essai Sur Le Pali, Ou Langue Sacree de La Presqu'ile Au-Dela Du Gange"). The first modern Pali-English dictionary was published by Robert Childers in 1872 and 1875. Following the foundation of the Pali Text Society, English Pali studies grew rapidly and Childer's dictionary became outdated. Planning for a new dictionary began in the early 1900s, but delays (including the outbreak of World War I) meant that work was not completed until 1925.


title: Pali

text: T. W. Rhys Davids in his book "Buddhist India", and Wilhelm Geiger in his book "Pāli Literature and Language", suggested that Pali may have originated as a lingua franca or common language of culture among people who used differing dialects in North India, used at the time of the Buddha and employed by him. Another scholar states that at that time it was "a refined and elegant vernacular of all Aryan-speaking people". Modern scholarship has not arrived at a consensus on the issue; there are a variety of conflicting theories with supporters and detractors. After the death of the Buddha, Pali may have evolved among Buddhists out of the language of the Buddha as a new artificial language. R. C. Childers, who held to the theory that Pali was Old Magadhi, wrote: "Had Gautama never preached, it is unlikely that Magadhese would have been distinguished from the many other vernaculars of Hindustan, except perhaps by an inherent grace and strength which make it a sort of Tuscan among the Prakrits."


title: Pali

text: According to K. R. Norman, differences between different texts within the canon suggest that it contains material from more than a single dialect. He also suggests it is likely that the viharas in North India had separate collections of material, preserved in the local dialect. In the early period it is likely that no degree of translation was necessary in communicating this material to other areas. Around the time of Ashoka there had been more linguistic divergence, and an attempt was made to assemble all the material. It is possible that a language quite close to the Pali of the canon emerged as a result of this process as a compromise of the various dialects in which the earliest material had been preserved, and this language functioned as a lingua franca among Eastern Buddhists from then on. Following this period, the language underwent a small degree of Sanskritisation (i.e., MIA bamhana > brahmana, tta > tva in some cases).


title: Pali

text: Bhikkhu Bodhi, summarizing the current state of scholarship, states that the language is "closely related to the language (or, more likely, the various regional dialects) that the Buddha himself spoke". He goes on to write:


title: Pali

text: According to A. K. Warder, the Pali language is a Prakrit language used in a region of Western India. Warder associates Pali with the Indian realm ("janapada") of Avanti, where the Sthavira nikāya was centered. Following the initial split in the Buddhist community, the Sthavira nikāya became influential in Western and South India while the Mahāsāṃghika branch became influential in Central and East India. Akira Hirakawa and Paul Groner also associate Pali with Western India and the Sthavira nikāya, citing the Saurashtran inscriptions, which are linguistically closest to the Pali language.


title: Pali

text: Although Sanskrit was said in the Brahmanical tradition to be the unchanging language spoken by the gods in which each word had an inherent significance, such views for any language was not shared in the early Buddhist traditions, in which words were only conventional and mutable signs. This view of language naturally extended to Pali and may have contributed to its usage (as an approximation or standardization of local Middle Indic dialects) in place of Sanskrit. However, by the time of the compilation of the Pali commentaries (4th or 5th century), Pali was described by the anonymous authors as the natural language, the root language of all beings.


title: Pali

text: Comparable to Ancient Egyptian, Latin or Hebrew in the mystic traditions of the West, Pali recitations were often thought to have a supernatural power (which could be attributed to their meaning, the character of the reciter, or the qualities of the language itself), and in the early strata of Buddhist literature we can already see Pali s used as charms, as, for example, against the bite of snakes. Many people in Theravada cultures still believe that taking a vow in Pali has a special significance, and, as one example of the supernatural power assigned to chanting in the language, the recitation of the vows of are believed to alleviate the pain of childbirth in Sri Lanka. In Thailand, the chanting of a portion of the is believed to be beneficial to the recently departed, and this ceremony routinely occupies as much as seven working days. There is nothing in the latter text that relates to this subject, and the origins of the custom are unclear.


title: Pali

text: Pali died out as a literary language in mainland India in the fourteenth century but survived elsewhere until the eighteenth. Today Pali is studied mainly to gain access to Buddhist scriptures, and is frequently chanted in a ritual context. The secular literature of Pali historical chronicles, medical texts, and inscriptions is also of great historical importance. The great centres of Pali learning remain in Sri Lanka and other Theravada nations of Southeast Asia: Myanmar, Thailand, Laos and Cambodia. Since the 19th century, various societies for the revival of Pali studies in India have promoted awareness of the language and its literature, including the Maha Bodhi Society founded by Anagarika Dhammapala.


title: Pali

text: In Europe, the Pali Text Society has been a major force in promoting the study of Pali by Western scholars since its founding in 1881. Based in the United Kingdom, the society publishes romanized Pali editions, along with many English translations of these sources. In 1869, the first "Pali Dictionary" was published using the research of Robert Caesar Childers, one of the founding members of the Pali Text Society. It was the first Pali translated text in English and was published in 1872. Childers' dictionary later received the Volney Prize in 1876.


title: Pali

text: The Pali Text Society was founded in part to compensate for the very low level of funds allocated to Indology in late 19th-century England and the rest of the UK; incongruously, the citizens of the UK were not nearly so robust in Sanskrit and Prakrit language studies as Germany, Russia, and even Denmark. Even without the inspiration of colonial holdings such as the former British occupation of Sri Lanka and Burma, institutions such as the Danish Royal Library have built up major collections of Pali manuscripts, and major traditions of Pali studies.


title: Pali

text: Pali literature is usually divided into canonical and non-canonical or extra-canonical texts. Canonical texts include the whole of the Pali Canon or "Tipitaka". With the exception of three books placed in the Khuddaka Nikaya by only the Burmese tradition, these texts (consisting of the five Nikayas of the Sutta Pitaka, the Vinaya Pitaka, and the books of the Abhidhamma Pitaka) are traditionally accepted as containing the words of the Buddha and his immediate disciples by the Theravada tradition.


title: Pali

text: Other types of texts present in Pali literature include works on grammar and poetics, medical texts, astrological and divination texts, cosmologies, and anthologies or collections of material from the canonical literature.


title: Pali

text: While the majority of works in Pali are believed to have originated with the Sri Lankan tradition and then spread to other Theravada regions, some texts may have other origins. The Milinda Panha may have originated in northern India before being translated from Sanskrit or Gandhari Prakrit. There are also a number of texts that are believed to have been composed in Pali in Sri Lanka, Thailand and Burma but were not widely circulated. This regional Pali literature is currently relatively little known, particularly in the Thai tradition, with many manuscripts never catalogued or published.


title: Pali

text: is a largely unattested literary language of classical India that is mentioned in Prakrit and Sanskrit grammars of antiquity. It is found grouped with the Prakrit languages, with which it shares some linguistic similarities, but was not considered a spoken language by the early grammarians because it was understood to have been purely a literary language.


title: Pali

text: In works of Sanskrit poetics such as Daṇḍin's "Kavyadarsha", it is also known by the name of , an epithet which can be interpreted as 'dead language' (i.e., with no surviving speakers), or means past and means language i.e. 'a language spoken in the past'. Evidence which lends support to this interpretation is that literature in Paiśācī is fragmentary and extremely rare but may once have been common.


title: Pali

text: The 13th-century Tibetan historian Buton Rinchen Drub wrote that the early Buddhist schools were separated by choice of sacred language: the Mahāsāṃghikas used Prākrit, the Sarvāstivādins used Sanskrit, the Sthaviravādins used Paiśācī, and the Saṃmitīya used Apabhraṃśa. This observation has led some scholars to theorize connections between Pali and Paiśācī; Sten Konow concluded that it may have been an Indo-Aryan language spoken by Dravidian people in South India, and Alfred Master noted a number of similarities between surviving fragments and Pali morphology.


title: Pali

text: Ardhamagadhi Prakrit was a Middle Indo-Aryan language and a Dramatic Prakrit thought to have been spoken in modern-day Bihar & Eastern Uttar Pradesh and used in some early Buddhist and Jain drama. It was originally thought to be a predecessor of the vernacular Magadhi Prakrit, hence the name (literally "half-Magadhi"). Ardhamāgadhī was prominently used by Jain scholars and is preserved in the Jain Agamas.


title: Pali

text: Ardhamagadhi Prakrit differs from later Magadhi Prakrit in similar ways to Pali, and was often believed to be connected with Pali on the basis of the belief that Pali recorded the speech of the Buddha in an early Magadhi dialect.


title: Pali

text: Magadhi Prakrit was a Middle Indic language spoken in present-day Bihar, and eastern Uttar Pradesh. Its use later expanded southeast to include some regions of modern-day Bengal, Odisha, and Assam, and it was used in some Prakrit dramas to represent vernacular dialogue. Preserved examples of Magadhi Prakrit are from several centuries after the theorized lifetime of the Buddha, and include inscriptions attributed to Asoka Maurya.


title: Pali

text: Differences observed between preserved examples of Magadhi Prakrit and Pali lead scholars to conclude that Pali represented a development of a northwestern dialect of Middle Indic, rather than being a continuation of a language spoken in the area of Magadha in the time of the Buddha.


title: Pali

text: Nearly every word in Pāḷi has cognates in the other Middle Indo-Aryan languages, the Prakrits. The relationship to Vedic Sanskrit is less direct and more complicated; the Prakrits were descended from Old Indo-Aryan vernaculars. Historically, influence between Pali and Sanskrit has been felt in both directions. The Pali language's resemblance to Sanskrit is often exaggerated by comparing it to later Sanskrit compositions—which were written centuries after Sanskrit ceased to be a living language, and are influenced by developments in Middle Indic, including the direct borrowing of a portion of the Middle Indic lexicon; whereas, a good deal of later Pali technical terminology has been borrowed from the vocabulary of equivalent disciplines in Sanskrit, either directly or with certain phonological adaptations.


title: Pali

text: Post-canonical Pali also possesses a few loan-words from local languages where Pali was used (e.g. Sri Lankans adding Sinhala words to Pali). These usages differentiate the Pali found in the from later compositions such as the Pali commentaries on the canon and folklore (e.g., commentaries on the Jataka tales), and comparative study (and dating) of texts on the basis of such loan-words is now a specialized field unto itself.


title: Pali

text: Pali was not exclusively used to convey the teachings of the Buddha, as can be deduced from the existence of a number of secular texts, such as books of medical science/instruction, in Pali. However, scholarly interest in the language has been focused upon religious and philosophical literature, because of the unique window it opens on one phase in the development of Buddhism.


title: Pali

text: Long and short vowels are only contrastive in open syllables; in closed syllables, all vowels are always short. Short and long e and o are in complementary distribution: the short variants occur only in closed syllables, the long variants occur only in open syllables. Short and long e and o are therefore not distinct phonemes.


title: Pali

text: when followed by a consonant with which they make a syllable as in [upek-khā] 'indifference', [sot-thi] 'safety'


title: Pali

text: A sound called "anusvāra" (Skt.; Pali: "niggahīta"), represented by the letter ' (ISO 15919) or ' (ALA-LC) in romanization, and by a raised dot in most traditional alphabets, originally marked the fact that the preceding vowel was nasalized. That is, ', ' and ' represented , and . In many traditional pronunciations, however, the anusvāra is pronounced more strongly, like the velar nasal , so that these sounds are pronounced instead , and . However pronounced, ' never follows a long vowel; ā, ī and ū are converted to the corresponding short vowels when is added to a stem ending in a long vowel, e.g. ' becomes ', not , ' becomes ', not *.


title: Pali

text: The final consonants of the Sanskrit words have been dropped in Pali and thus all the words end in a vowel or in a nasal vowel: " -> kantā 'from the loved one"; " -> 'the loved one"


title: Pali

text: The final vowels were usually weak in pronunciation and hence they were shortened: "akārsit -> akāsi 'he did'."


title: Pali

text: Among the labial consonants, is labiodental and the rest is bilabial. Among the dental/alveolar consonants, the majority is dental but and are alveolar.


title: Pali

text: Of the sounds listed above only the three consonants in parentheses, ṅ, ḷ, and ḷh, are not distinct phonemes in Pali: ṅ only occurs before velar stops, while ḷ and ḷh are allophones of single ḍ and ḍh occurring between vowels.


title: Pali

text: In the Pali language, the consonants may be divided according to their strength or power of resistance. The strength decreases in the order of: "mutes, sibilant, nasals, l, v, y, r"


title: Pali

text: when one of the two consonants is the sibilant s, then the new group of consonants has the aspiration in the last consonant: "as-ti (root: √as) > atthi" 'is'


title: Pali

text: the sibilant s, followed by a nasal, is changed to h and then it is transposed after the nasal (metathesis): "akas-ma > akah-ma > akamha" 'we did'


title: Pali

text: Pali v appears for Skr. y. For instance, "āvudha -> āyudha" 'weapon'; "kasāva -> kasāya" 'dirt, sin'. After the svarabhakti-vowel I there appear v instead of y as in "praṭyamsa -> pativimsa."


title: Pali

text: Representation of "r" by "l" is very common in Pali, and in Pkr. it is the rule for Magadhi, although this substitution occurs sporadically also in other dialect. This, initially, in "lūjjati -> rūjyate 'falls apart"; sometimes double forms with l and r occur in Skr.: "lūkha -> lūksa, rūksa 'gross, bad"


title: Pali

text: Pali is a highly inflected language, in which almost every word contains, besides the root conveying the basic meaning, one or more affixes (usually suffixes) which modify the meaning in some way. Nouns are inflected for gender, number, and case; verbal inflections convey information about person, number, tense and mood.


title: Pali

text: Pali nouns inflect for three grammatical genders (masculine, feminine, and neuter) and two numbers (singular and plural). The nouns also, in principle, display eight cases: nominative or "paccatta" case, vocative, accusative or "upayoga" case, instrumental or "" case, dative or "sampadāna" case, ablative, genitive or "sāmin" case, and locative or "bhumma" case; however, in many instances, two or more of these cases are identical in form; this is especially true of the genitive and dative cases.


title: Pali

text: a-stems, whose uninflected stem ends in short "a" (), are either masculine or neuter. The masculine and neuter forms differ only in the nominative, vocative, and accusative cases.


title: Pali

text: i-stems and u-stems are either masculine or neuter. The masculine and neuter forms differ only in the nominative and accusative cases. The vocative has the same form as the nominative.


title: Pali

text: The literal meaning is therefore: "The dharmas have mind as their leader, mind as their chief, are made of/by mind. If [someone] either speaks or acts with a corrupted mind, from that [cause] suffering goes after him, as the wheel [of a cart follows] the foot of a draught animal."


title: Pali

text: Pali and Sanskrit are very closely related and the common characteristics of Pali and Sanskrit were always easily recognized by those in India who were familiar with both. A large part of Pali and Sanskrit word-stems are identical in form, differing only in details of inflection.


title: Pali

text: Technical terms from Sanskrit were converted into Pali by a set of conventional phonological transformations. These transformations mimicked a subset of the phonological developments that had occurred in Proto-Pali. Because of the prevalence of these transformations, it is not always possible to tell whether a given Pali word is a part of the old Prakrit lexicon, or a transformed borrowing from Sanskrit. The existence of a Sanskrit word regularly corresponding to a Pali word is not always secure evidence of the Pali etymology, since, in some cases, artificial Sanskrit words were created by back-formation from Prakrit words.


title: Pali

text: The following phonological processes are not intended as an exhaustive description of the historical changes which produced Pali from its Old Indic ancestor, but rather are a summary of the most common phonological equations between Sanskrit and Pali, with no claim to completeness.


title: Pali

text: Total assimilation, where one sound becomes identical to a neighboring sound, is of two types: progressive, where the assimilated sound becomes identical to the following sound; and regressive, where it becomes identical to the preceding sound.


title: Pali

text: An epenthetic vowel is sometimes inserted between certain consonant-sequences. As with ṛ, the vowel may be a, i, or u, depending on the influence of a neighboring consonant or of the vowel in the following syllable. i is often found near i, y, or palatal consonants; u is found near u, v, or labial consonants.


title: Pali

text: There are several notable exceptions to the rules above; many of them are common Prakrit words rather than borrowings from Sanskrit.


title: Pali

text: Emperor Ashoka erected a number of pillars with his edicts in at least three regional Prakrit languages in Brahmi script, all of which are quite similar to Pali. Historically, the first written record of the Pali canon is believed to have been composed in Sri Lanka, based on a prior oral tradition. According to the Mahavamsa (the chronicle of Sri Lanka), due to a major famine in the country Buddhist monks wrote down the Pali canon during the time of King Vattagamini in 100 BCE.


title: Pali

text: Bilingual coins containing Pali written in the Kharosthi script and Greek writing were used by James Prinsep to decipher the Kharosthi abugida. This script became particularly significant for the study of early Buddhism following the discovery of the Gandharan Buddhist texts.


title: Pali

text: The transmission of written Pali has retained a universal system of alphabetic values, but has expressed those values in a variety of different scripts.


title: Pali

text: In Sri Lanka, Pali texts were recorded in Sinhala script. Other local scripts, most prominently Khmer, Mon-Burmese, and in modern times Thai (since 1893) and Devanāgarī have been used to record Pali.


title: Pali

text: Since the 19th century, Pali has also been written in the Roman script. An alternate scheme devised by Frans Velthuis, called the Velthuis scheme (see § Text in ASCII) allows for typing without diacritics using plain ASCII methods, but is arguably less readable than the standard IAST system, which uses diacritical marks.


title: Pali

text: There are several fonts to use for Pali transliteration. However, older ASCII fonts such as Leedsbit PaliTranslit, Times_Norman, Times_CSX+, Skt Times, Vri RomanPali CN/CB etc., are not recommendable, they are deprecated, since they are not compatible with one another, and are technically out of date. Instead, fonts based on the Unicode standard are recommended.


title: Pali

text: However, not all Unicode fonts contain the necessary characters. To properly display all the diacritic marks used for romanized Pali (or for that matter, Sanskrit), a Unicode font must contain the following character ranges:


title: Pali

text: Some of the latest fonts coming with Windows 7 can also be used to type transliterated Pali: "Arial", "Calibri", "Cambria", "Courier New", "Microsoft Sans Serif", "Segoe UI", "Segoe UI Light", "Segoe UI Semibold", "Tahoma", and "Times New Roman". Some of them have four styles each, hence usable in professional typesetting: "Arial, Calibri" and "Segoe UI" are sans-serif fonts, "Cambria" and "Times New Roman" are serif fonts and "Courier New" is a monospace font.


title: Pali

text: The Velthuis scheme was originally developed in 1991 by Frans Velthuis for use with his "devnag" Devanāgarī font, designed for the TeX typesetting system. This system of representing Pali diacritical marks has been used in some websites and discussion lists. However, as the Web itself and email software slowly evolve towards the Unicode encoding standard, this system has become almost unnecessary and obsolete.


title: Pythagorean triple

text: A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.


title: Pythagorean triple

text: The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula formula_1; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides formula_2 and formula_3 is a right triangle, but formula_4 is not a Pythagorean triple because formula_5 is not an integer. Moreover, formula_6 and formula_5 do not have an integer common multiple because formula_5 is irrational.


title: Pythagorean triple

text: Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10.


title: Pythagorean triple

text: When searching for integer solutions, the equation is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation.


title: Pythagorean triple

text: Other small Pythagorean triples such as (6, 8, 10) are not listed because they are not primitive; for instance (6, 8, 10) is a multiple of (3, 4, 5).


title: Pythagorean triple

text: Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers and with . The formula states that the integers


title: Pythagorean triple

text: form a Pythagorean triple. The triple generated by Euclid's formula is primitive if and only if and are coprime and one of them is even. When both and are odd, then , , and will be even, and the triple will not be primitive; however, dividing , , and by 2 will yield a primitive triple when and are coprime.


title: Pythagorean triple

text: "Every" primitive triple arises (after the exchange of and , if is even) from a "unique pair" of coprime numbers , , one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of , and to and from Euclid's formula is referenced throughout the rest of this article.


title: Pythagorean triple

text: Despite generating all primitive triples, Euclid's formula does not produce all triples—for example, (9, 12, 15) cannot be generated using integer and . This can be remedied by inserting an additional parameter to the formula. The following will generate all Pythagorean triples uniquely:


title: Pythagorean triple

text: That these formulas generate Pythagorean triples can be verified by expanding using elementary algebra and verifying that the result equals . Since every Pythagorean triple can be divided through by some integer to obtain a primitive triple, every triple can be generated uniquely by using the formula with and to generate its primitive counterpart and then multiplying through by as in the last equation.


title: Pythagorean triple

text: Choosing and from certain integer sequences gives interesting results. For example, if and are consecutive Pell numbers, and will differ by 1.


title: Pythagorean triple

text: Many formulas for generating triples with particular properties have been developed since the time of Euclid.


title: Pythagorean triple

text: That satisfaction of Euclid's formula by "a, b, c" is sufficient for the triangle to be Pythagorean is apparent from the fact that for positive integers and , , the , , and given by the formula are all positive integers, and from the fact that


title: Pythagorean triple

text: A proof of the "necessity" that "a, b, c" be expressed by Euclid's formula for any primitive Pythagorean triple is as follows. All such primitive triples can be written as where and , , are coprime. Thus , , are pairwise coprime (if a prime number divided two of them, it would be forced also to divide the third one). As and are coprime, at least one of them is odd, so we may suppose that is odd, by exchanging, if needed, and . This implies that is even and is odd (if were odd, would be even, and would be a multiple of 4, while would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4).


title: Pythagorean triple

text: From formula_1 we obtain formula_13 and hence formula_14. Then formula_15. Since formula_16 is rational, we set it equal to formula_17 in lowest terms. Thus formula_18, being the reciprocal of formula_16. Then solving


title: Pythagorean triple

text: As formula_17 is fully reduced, and are coprime, and they cannot both be even. If they were both odd, the numerator of formula_25 would be a multiple of 4 (because an odd square is congruent to 1 modulo 4), and the denominator 2"mn" would not be a multiple of 4. Since 4 would be the minimum possible even factor in the numerator and 2 would be the maximum possible even factor in the denominator, this would imply to be even despite defining it as odd. Thus one of and is odd and the other is even, and the numerators of the two fractions with denominator 2"mn" are odd. Thus these fractions are fully reduced (an odd prime dividing this denominator divides one of and but not the other; thus it does not divide ). One may thus equate numerators with numerators and denominators with denominators, giving Euclid's formula


title: Pythagorean triple

text: A longer but more commonplace proof is given in Maor (2007) and Sierpiński (2003). Another proof is given in , as an instance of a general method that applies to every homogeneous Diophantine equation of degree two.


title: Pythagorean triple

text: Suppose the sides of a Pythagorean triangle have lengths , , and , and suppose the angle between the leg of length and the hypotenuse of length is denoted as . Then formula_27 and the full-angle trigonometric values are formula_28, formula_29, and formula_30.


title: Pythagorean triple

text: The following variant of Euclid's formula is sometimes more convenient, as being more symmetric in and (same parity condition on and ).


title: Pythagorean triple

text: are three integers that form a Pythagorean triple, which is primitive if and only if and are coprime. Conversely, every primitive Pythagorean triple arises (after the exchange of and , if is even) from a unique pair of coprime odd integers.


title: Pythagorean triple

text: The properties of a primitive Pythagorean triple with (without specifying which of or is even and which is odd) include:


title: Pythagorean triple

text: In addition, special Pythagorean triples with certain additional properties can be guaranteed to exist:


title: Pythagorean triple

text: In fact, a point in the Cartesian plane with coordinates belongs to the unit circle if . The point is "rational" if and are rational numbers, that is, if there are coprime integers such that


title: Pythagorean triple

text: By multiplying both members by , one can see that the rational points on the circle are in one-to-one correspondence with the primitive Pythagorean triples.


title: Pythagorean triple

text: Euclid's formula for Pythagorean triples and the inverse relationship mean that, except for , a point on the circle is rational if and only if the corresponding value of is a rational number. Note that is also the tangent of half of the angle that is opposite the triangle side of length .


title: Pythagorean triple

text: There is a correspondence between points on the unit circle with rational coordinates and primitive Pythagorean triples. At this point, Euclid's formulae can be derived either by methods of trigonometry or equivalently by using the stereographic projection.


title: Pythagorean triple

text: This establishes that each rational point of the -axis goes over to a rational point of the unit circle. The converse, that every rational point of the unit circle comes from such a point of the -axis, follows by applying the inverse stereographic projection. Suppose that is a point of the unit circle with and rational numbers. Then the point ′ obtained by stereographic projection onto the -axis has coordinates


title: Pythagorean triple

text: In terms of algebraic geometry, the algebraic variety of rational points on the unit circle is birational to the affine line over the rational numbers. The unit circle is thus called a rational curve, and it is this fact which enables an explicit parameterization of the (rational number) points on it by means of rational functions.


title: Pythagorean triple

text: A 2D lattice is a regular array of isolated points where if any one point is chosen as the Cartesian origin (0, 0), then all the other points are at where and range over all positive and negative integers. Any Pythagorean triangle with triple can be drawn within a 2D lattice with vertices at coordinates , and . The count of lattice points lying strictly within the bounds of the triangle is given by   formula_48 for primitive Pythagorean triples this interior lattice count is  formula_49 The area (by Pick's theorem equal to one less than the interior lattice count plus half the boundary lattice count) equals  formula_50 .


title: Pythagorean triple

text: The first occurrence of two primitive Pythagorean triples sharing the same area occurs with triangles with sides and common area 210 . The first occurrence of two primitive Pythagorean triples sharing the same interior lattice count occurs with and interior lattice count 2287674594 . Three primitive Pythagorean triples have been found sharing the same area: , , with area 13123110. As yet, no set of three primitive Pythagorean triples have been found sharing the same interior lattice count.


title: Pythagorean triple

text: By Euclid's formula all primitive Pythagorean triples can be generated from integers formula_51 and formula_52 with formula_53, formula_54 odd and formula_55. Hence there is a 1 to 1 mapping of rationals (in lowest terms) to primitive Pythagorean triples where formula_56 is in the interval formula_57 and formula_54 odd.


title: Pythagorean triple

text: The reverse mapping from a primitive triple formula_59 where formula_60 to a rational formula_56 is achieved by studying the two sums formula_62 and formula_63. One of these sums will be a square that can be equated to formula_64 and the other will be twice a square that can be equated to formula_65. It is then possible to determine the rational formula_56.


title: Pythagorean triple

text: In order to enumerate primitive Pythagorean triples the rational can be expressed as an ordered pair formula_67 and mapped to an integer using a pairing function such as Cantor's pairing function. An example can be seen at . It begins


title: Pythagorean triple

text: which is zero precisely when is a Pythagorean triple. If corresponds to a Pythagorean triple, then as a matrix it must have rank 1.


title: Pythagorean triple

text: Since is symmetric, it follows from a result in linear algebra that there is a column vector such that the outer product


title: Pythagorean triple

text: holds, where the denotes the matrix transpose. The vector ξ is called a spinor (for the Lorentz group SO(1, 2)). In abstract terms, the Euclid formula means that each primitive Pythagorean triple can be written as the outer product with itself of a spinor with integer entries, as in ().


title: Pythagorean triple

text: with determinant equal to one: . This set forms a group, since the inverse of a matrix in Γ is again in Γ, as is the product of two matrices in Γ. The modular group acts on the collection of all integer spinors. Furthermore, the group is transitive on the collection of integer spinors with relatively prime entries. For if has relatively prime entries, then


title: Pythagorean triple

text: By acting on the spinor ξ in (), the action of Γ goes over to an action on Pythagorean triples, provided one allows for triples with possibly negative components. Thus if is a matrix in , then


title: Pythagorean triple

text: gives rise to an action on the matrix in (). This does not give a well-defined action on primitive triples, since it may take a primitive triple to an imprimitive one. It is convenient at this point (per ) to call a triple standard if and either are relatively prime or are relatively prime with odd. If the spinor has relatively prime entries, then the associated triple determined by () is a standard triple. It follows that the action of the modular group is transitive on the set of standard triples.


title: Pythagorean triple

text: Alternatively, restrict attention to those values of and for which is odd and is even. Let the subgroup Γ(2) of Γ be the kernel of the group homomorphism


title: Pythagorean triple

text: where is the special linear group over the finite field of integers modulo 2. Then Γ(2) is the group of unimodular transformations which preserve the parity of each entry. Thus if the first entry of ξ is odd and the second entry is even, then the same is true of for all . In fact, under the action (), the group Γ(2) acts transitively on the collection of primitive Pythagorean triples .


title: Pythagorean triple

text: Consequently, every primitive Pythagorean triple can be obtained in a unique way as a product of copies of the matrices and .


title: Pythagorean triple

text: By a result of , all primitive Pythagorean triples can be generated from the (3, 4, 5) triangle by using the three linear transformations T, T, T below, where , , are sides of a triple:


title: Pythagorean triple

text: The linear transformations T, T, and T have a geometric interpretation in the language of quadratic forms. They are closely related to (but are not equal to) reflections generating the orthogonal group of over the integers.


title: Pythagorean triple

text: Alternatively, Euclid's formulae can be analyzed and proved using the Gaussian integers. Gaussian integers are complex numbers of the form , where and are ordinary integers and is the square root of negative one. The units of Gaussian integers are ±1 and ±i. The ordinary integers are called the rational integers and denoted as ". The Gaussian integers are denoted as . The right-hand side of the Pythagorean theorem may be factored in Gaussian integers:


title: Pythagorean triple

text: A primitive Pythagorean triple is one in which and are coprime, i.e., they share no prime factors in the integers. For such a triple, either or is even, and the other is odd; from this, it follows that is also odd.


title: Pythagorean triple

text: The two factors and of a primitive Pythagorean triple each equal the square of a Gaussian integer. This can be proved using the property that every Gaussian integer can be factored uniquely into Gaussian primes up to units. (This unique factorization follows from the fact that, roughly speaking, a version of the Euclidean algorithm can be defined on them.) The proof has three steps. First, if and share no prime factors in the integers, then they also share no prime factors in the Gaussian integers. (Assume and with Gaussian integers , and and not a unit. Then and lie on the same line through the origin. All Gaussian integers on such a line are integer multiples of some Gaussian integer . But then the integer "gh" ≠ ±1 divides both and .) Second, it follows that and likewise share no prime factors in the Gaussian integers. For if they did, then their common divisor would also divide and . Since and are coprime, that implies that divides . From the formula , that in turn would imply that is even, contrary to the hypothesis of a primitive Pythagorean triple. Third, since is a square, every Gaussian prime in its factorization is doubled, i.e., appears an even number of times. Since and share no prime factors, this doubling is also true for them. Hence, and are squares.


title: Pythagorean triple

text: For any primitive Pythagorean triple, there must be integers and such that these two equations are satisfied. Hence, every Pythagorean triple can be generated from some choice of these integers.


title: Pythagorean triple

text: If we consider the square of a Gaussian integer we get the following direct interpretation of Euclid's formula as representing the perfect square of a Gaussian integer.


title: Pythagorean triple

text: Using the facts that the Gaussian integers are a Euclidean domain and that for a Gaussian integer p formula_81 is always a square it is possible to show that a Pythagorean triple corresponds to the square of a prime Gaussian integer if the hypotenuse is prime.


title: Pythagorean triple

text: If the Gaussian integer is not prime then it is the product of two Gaussian integers p and q with formula_81 and formula_83 integers. Since magnitudes multiply in the Gaussian integers, the product must be formula_84, which when squared to find a Pythagorean triple must be composite. The contrapositive completes the proof.


title: Pythagorean triple

text: There are a number of results on the distribution of Pythagorean triples. In the scatter plot, a number of obvious patterns are already apparent. Whenever the legs of a primitive triple appear in the plot, all integer multiples of must also appear in the plot, and this property produces the appearance of lines radiating from the origin in the diagram.


title: Pythagorean triple

text: Within the scatter, there are sets of parabolic patterns with a high density of points and all their foci at the origin, opening up in all four directions. Different parabolas intersect at the axes and appear to reflect off the axis with an incidence angle of 45 degrees, with a third parabola entering in a perpendicular fashion. Within this quadrant, each arc centered on the origin shows that section of the parabola that lies between its tip and its intersection with its semi-latus rectum.


title: Pythagorean triple

text: These patterns can be explained as follows. If formula_85 is an integer, then (, formula_86, formula_87) is a Pythagorean triple. (In fact every Pythagorean triple can be written in this way with integer , possibly after exchanging and , since formula_88 and and cannot both be odd.) The Pythagorean triples thus lie on curves given by formula_89, that is, parabolas reflected at the -axis, and the corresponding curves with and interchanged. If is varied for a given (i.e. on a given parabola), integer values of occur relatively frequently if is a square or a small multiple of a square. If several such values happen to lie close together, the corresponding parabolas approximately coincide, and the triples cluster in a narrow parabolic strip. For instance, , ,


title: Pythagorean triple

text: The angular properties described above follow immediately from the functional form of the parabolas. The parabolas are reflected at the -axis at , and the derivative of with respect to at this point is –1; hence the incidence angle is 45°. Since the clusters, like all triples, are repeated at integer multiples, the value also corresponds to a cluster. The corresponding parabola intersects the -axis at right angles at , and hence its reflection upon interchange of and intersects the -axis at right angles at , precisely where the parabola for is reflected at the -axis. (The same is of course true for and interchanged.)


title: Pythagorean triple

text: Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.


title: Pythagorean triple

text: The case of the more general construction of Pythagorean triples has been known for a long time. Proclus, in his commentary to the 47th Proposition of the first book of Euclid's "Elements", describes it as follows:


title: Pythagorean triple

text: Certain methods for the discovery of triangles of this kind are handed down, one which they refer to Plato, and another to Pythagoras. (The latter) starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side, the hypotenuse.


title: Pythagorean triple

text: ...For the method of Plato argues from even numbers. It takes the given even number and makes it one of the sides about the right angle; then, bisecting this number and squaring the half, it adds unity to the square to form the hypotenuse, and subtracts unity from the square to form the other side about the right angle. ... Thus it has formed the same triangle that which was obtained by the other method.


title: Pythagorean triple

text: It can be shown that all Pythagorean triples can be obtained, with appropriate rescaling, from the basic Platonic sequence (, and ) by allowing to take non-integer rational values. If is replaced with the fraction in the sequence, the result is equal to the 'standard' triple generator (2"mn", ,) after rescaling. It follows that every triple has a corresponding rational value which can be used to generate a similar triangle (one with the same three angles and with sides in the same proportions as the original). For example, the Platonic equivalent of is generated by as . The Platonic sequence itself can be derived by following the steps for 'splitting the square' described in Diophantus II.VIII.


title: Pythagorean triple

text: There is an infinite number of solutions to this equation as solving for the variables involves an elliptic curve. Small ones are,


title: Pythagorean triple

text: One way to generate solutions to formula_96 is to parametrize "a, b, c, d" in terms of integers "m, n, p, q" as follows:


title: Pythagorean triple

text: and was first solved by Euler as formula_102. Since he showed this is a rational point in an elliptic curve, then there is an infinite number of solutions. In fact, he also found a 7th degree polynomial parameterization.


title: Pythagorean triple

text: There is also an infinite number of solutions, and for the special case when formula_107, then the equation simplifies to,


title: Pythagorean triple

text: No Pythagorean triples are isosceles, because the ratio of the hypotenuse to either other side is , but cannot be expressed as the ratio of 2 integers.


title: Pythagorean triple

text: There are, however, right-angled triangles with integral sides for which the lengths of the non-hypotenuse sides differ by one, such as,


title: Pythagorean triple

text: If , , are the sides of this type of primitive Pythagorean triple (PPT) then the solution to the Pell equation is given by the recursive formula


title: Pythagorean triple

text: It shows that all odd numbers (greater than 1) appear in this type of almost-isosceles PPT. This sequence of PPTs forms the right hand side outer stem of the rooted ternary tree of PPTs.


title: Pythagorean triple

text: Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple, obtained from the formula


title: Pythagorean triple

text: is a Pythagorean -tuple for any tuple of positive integers with . The Pythagorean -tuple can be made primitive by dividing out by the largest common divisor of its values.


title: Pythagorean triple

text: Furthermore, any primitive Pythagorean -tuple can be found by this approach. Use to get a Pythagorean -tuple by the above formula and divide out by the largest common integer divisor, which is . Dividing out by the largest common divisor of these values gives the same primitive Pythagorean -tuple; and there is a one-to-one correspondence between tuples of setwise coprime positive integers satisfying and primitive Pythagorean -tuples.


title: Pythagorean triple

text: Examples of the relationship between setwise coprime values formula_135 and primitive Pythagorean -tuples include:


title: Pythagorean triple

text: one may find values so that is a square, such as one by Hirschhorn where the number of terms is itself a square,


title: Pythagorean triple

text: and is any integer not divisible by 2 or 3. For the smallest case , hence , this yields the well-known cannonball-stacking problem of Lucas,


title: Pythagorean triple

text: In addition, if in a Pythagorean -tuple () all addends are consecutive except one, one can use the equation,


title: Pythagorean triple

text: Since the second power of cancels out, this is only linear and easily solved for as formula_141 though , should be chosen so that is an integer, with a small example being , yielding,


title: Pythagorean triple

text: A generalization of the concept of Pythagorean triples is the search for triples of positive integers , , and , such that , for some strictly greater than 2. Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as Fermat's Last Theorem because it took longer than any other conjecture by Fermat to be proved or disproved. The first proof was given by Andrew Wiles in 1994.


title: Pythagorean triple

text: Another generalization is searching for sequences of positive integers for which the th power of the last is the sum of the th powers of the previous terms. The smallest sequences for known values of are:


title: Pythagorean triple

text: For the case, in which formula_145 called the Fermat cubic, a general formula exists giving all solutions.


title: Pythagorean triple

text: A slightly different generalization allows the sum of th powers to equal the sum of th powers. For example:


title: Pythagorean triple

text: There can also exist positive integers whose th powers sum to an th power (though, by Fermat's Last Theorem, not for ; these are counterexamples to Euler's sum of powers conjecture. The smallest known counterexamples are


title: Pythagorean triple

text: A Heronian triangle is commonly defined as one with integer sides whose area is also an integer. The lengths of the sides of such a triangle form a Heronian triple for .


title: Pythagorean triple

text: Every Pythagorean triple is a Heronian triple, because at least one of the legs , must be even in a Pythagorean triple, so the area "ab"/2 is an integer. Not every Heronian triple is a Pythagorean triple, however, as the example with area 24 shows.


title: Pythagorean triple

text: If is a Heronian triple, so is where is any positive integer; its area will be the integer that is times the integer area of the triangle.


title: Pythagorean triple

text: The Heronian triple is primitive provided "a", "b", "c" are setwise coprime. (With primitive Pythagorean triples the stronger statement that they are "pairwise" coprime also applies, but with primitive Heronian triangles the stronger statement does not always hold true, such as with .) Here are a few of the simplest primitive Heronian triples that are not Pythagorean triples:


title: Pythagorean triple

text: By Heron's formula, the extra condition for a triple of positive integers with to be Heronian is that


title: Pythagorean triple

text: Primitive Pythagorean triples have been used in cryptography as random sequences and for the generation of keys.


title: Attack aircraft

text: An attack aircraft, strike aircraft, or attack bomber is a tactical military aircraft that has a primary role of carrying out airstrikes with greater precision than bombers, and is prepared to encounter strong low-level air defenses while pressing the attack. This class of aircraft is designed mostly for close air support and naval air-to-surface missions, overlapping the tactical bomber mission. Designs dedicated to non-naval roles are often known as ground-attack aircraft.


title: Attack aircraft

text: Fighter aircraft often carry out the attack role, although they would not be considered attack aircraft "per se", although fighter-bomber conversions of those same aircraft would be considered part of the class. Strike fighters, which have effectively replaced the fighter-bomber and light bomber concepts, also differ little from the broad concept of an attack aircraft.


title: Attack aircraft

text: The dedicated attack aircraft as a separate class existed primarily during and after World War II. The precise implementation varied from country to country, and was handled by a wide variety of designs. In the United States and Britain, attack aircraft were generally light bombers or medium bombers, sometimes carrying heavier forward-firing weapons like the North American B-25G Mitchell and de Havilland Mosquito Tsetse. In Germany and the USSR, where they were known as "Schlachtflugzeug" ("battle aircraft") or "sturmovik" ("storm trooper") respectively, this role was carried out by purpose-designed and heavily armored aircraft such as the Henschel Hs 129 and Ilyushin Il-2. The Germans and Soviets also used light bombers in this role: cannon-armed versions of the Junkers Ju 87 Stuka greatly outnumbered the Hs 129, while the Petlyakov Pe-2 was used for this role in spite of not being specifically designed for it.


title: Attack aircraft

text: In the latter part of World War II the fighter-bomber began to take over many attack roles, a transition that continued in the post-war era. Jet-powered examples were relatively rare but not unknown, such as the Blackburn Buccaneer. The U.S. Navy continued to introduce new aircraft in their A-series, but these were mostly similar to light and medium bombers. The need for a separate attack aircraft category was greatly diminished by the introduction of precision-guided munitions which allowed almost any aircraft to carry out this role while remaining safe at high altitude. Attack helicopters also have overtaken many remaining roles that could only be carried out at lower altitudes.


title: Attack aircraft

text: Since the 1960s, only two dedicated attack aircraft designs have been widely introduced, the American Fairchild Republic A-10 Thunderbolt II and Soviet/Russian Sukhoi Su-25 Frogfoot. The only anomaly belonging to this class of attack aircraft is the American Lockheed AC-130, which features its primary armament of artillery guns adapted for aircraft use, including the 105 mm M102 howitzer.


title: Attack aircraft

text: A variety of light attack aircraft has also been introduced in the post-World War II era, usually based on adapted trainers or other light fixed-wing aircraft. These have been used in counter-insurgency operations.


title: Attack aircraft

text: U.S. attack aircraft are currently identified by the prefix A-, as in "A-6 Intruder" and "A-10 Thunderbolt II". However, until the end of World War II the "A-" designation was shared between attack planes and light bombers for USAAF aircraft (as opposed to "B-" prefix for medium or heavy bombers). The US Navy used a separate designation system and at the time preferred to call similar aircraft scout bombers (SB) or torpedo bombers (TB or BT). For example, Douglas SBD Dauntless scout bomber was designated A-24 when used by the USAAF. It was not until 1946, when the US Navy and US Marine Corps started using the "attack" (A) designation, when it renamed BT2D Skyraider and BTM Mauler to, respectively, AD Skyraider and AM Mauler.


title: Attack aircraft

text: As with many aircraft classifications, the definition of "attack aircraft" is somewhat vague and has tended to change over time. Current U.S. military doctrine defines it as an aircraft which most likely performs an "attack mission", more than any other kind of mission. "Attack mission" means, in turn, specifically tactical air-to-ground action—in other words, neither air-to-air action nor strategic bombing is considered an "attack mission". In United States Navy vocabulary, the alternative designation for the same activity is a "strike mission". Attack missions are principally divided into two categories: air interdiction and close air support. In the last several decades, the rise of the ubiquitous multi-role fighter has created some confusion about the difference between attack and fighter aircraft. According to the current U.S. designation system, an attack aircraft ("A") is designed primarily for air-to-surface (Attack: Aircraft designed to find, attack, and destroy land or sea targets) missions (also known as "attack missions"), while a fighter category "F" incorporates not only aircraft designed primarily for air-to-air combat, but additionally multipurpose aircraft designed also for ground-attack missions. "F - Fighter Aircraft were designed to intercept and destroy other aircraft or missiles. This includes multipurpose aircraft also designed for ground support missions such as interdiction and close air support. Just to mention one example amongst many, the F-111 "Aardvark" was designated "F" despite having only minimal air-to-air capabilities. Only a single aircraft in the USAF's current inventory bears a simple, unmixed "A" designation: the A-10 Thunderbolt II.


title: Attack aircraft

text: British designations have included FB for fighter-bomber and more recently "G" for "Ground-attack" as in Harrier GR1 (meaning "Ground-attack/Reconnaissance, Mark 1").


title: Attack aircraft

text: Imperial Japanese Navy designation use "B" to designate carrier attack bomber such as the Nakajima B5N Type-97 bomber although these aircraft are mostly used for torpedo attack and level bombing. They also use "D" to specifically designate carrier dive bomber like the Yokosuka D4Y "Suisei""." However by the end of the world war 2, the IJN introduced the Aichi B7A "Ryusei" which could performed both torpedo bombing and dive bombing rendering the "D" designation redundant.


title: Attack aircraft

text: The NATO reporting names for Soviet/Russian ground-attack aircraft at first started with "B" categorizing them as bombers, as in case of Il-10 'Beast'. But later they were usually classified as fighters ("F")—possibly because (since Sukhoi Su-7) they were similar in size and visual appearance to Soviet fighters, or were simply derivatives of such.


title: Attack aircraft

text: The attack aircraft as a role was defined by its use during World War I, in support of ground forces on battlefields. Battlefield support is generally divided into close air support and battlefield air interdiction, the first requiring strict and the latter only general cooperation with friendly surface forces. Such aircraft also attacked targets in rear areas. Such missions required flying where light anti-aircraft fire was expected and operating at low altitudes to precisely identify targets. Other roles, including those of light bombers, medium bombers, dive bombers, reconnaissance, fighters, fighter-bombers, could and did perform air strikes on battlefields. All these types could significantly damage ground targets from a low level flight, either by bombing, machine guns, or both.


title: Attack aircraft

text: Attack aircraft came to diverge from bombers and fighters. While bombers could be used on a battlefield, their slower speeds made them extremely vulnerable to ground fire, as did the lighter construction of fighters. The survivability of attack aircraft was guaranteed by their speed/power, protection (i.e. armor panels) and strength of construction;


title: Attack aircraft

text: Germany was the first country to produce dedicated ground-attack aircraft (designated CL-class and J-class). They were put into use in autumn 1917, during World War I. Most notable was the Junkers J.I, which pioneered the idea of an armored "bathtub", that was both fuselage structure and protection for engine and crew. The British experimented with the Sopwith TF series (termed "trench fighters"), although these did not see combat.


title: Attack aircraft

text: The last battles of 1918 on the Western Front demonstrated that ground-attacking aircraft were a valuable component of all-arms tactics. Close support ground strafing (machine-gunning) and tactical bombing of infantry (especially when moving between trenches and along roads), machine gun posts, artillery, and supply formations was a part of the Allied armies' strength in holding German attacks and supporting Allied counter-attacks and offensives. Admittedly, the cost to the Allies was high, with the Royal Flying Corps sustaining a loss rate approaching 30% among ground-attack aircraft.


title: Attack aircraft

text: After World War I, it was widely believed that using aircraft against tactical targets was of little use other than in harassing and undermining enemy morale; attacking combatants was generally much more dangerous to aircrews than their targets, a problem that was continually becoming more acute with the ongoing refinement of anti-aircraft weapons. Within the range of types serving attack roles, dive bombers were increasingly being seen as more effective than aircraft designed for strafing with machine guns or cannons.


title: Attack aircraft

text: Nevertheless, during the 1920s, the US military, in particular, procured specialized "Attack" aircraft and formed dedicated units, that were trained primarily for that role. The US Army Engineering Division became involved in designing ground attack aircraft. The 1920 Boeing GA-1 was an armored twin-engine triplane for ground strafing with eight machine guns and about a ton of armor plate, and the 1922 Aeromarine PG-1 was a combined pursuit (fighter) and ground attack design with a 37mm gun. The United States Marine Corps Aviation applied close air support tactics in the Banana Wars. While they did not pioneer dive bombing tactics, Marine aviators were the first to include it in their doctrine during the United States occupation of Haiti and Nicaragua. The United States Army Air Corps was notable for its creation of a separate "A-" designation for attack types, distinct from and alongside "B-" for bomber types and "P-" for pursuit (later replaced by "F-" for fighter) aircraft. The first designated attack type to be operational with the USAAC was the Curtiss A-2 Falcon. Nevertheless, such aircraft, including the A-2's replacement, the Curtiss A-12 Shrike, were unarmored and highly vulnerable to AA fire.


title: Attack aircraft

text: The British Royal Air Force focused primarily on strategic bombing, rather than ground attack. However, like most air arms of the period it did operate attack aircraft, named Army Cooperation in RAF parlance, which included the Hawker Hector, Westland Lysander and others.


title: Attack aircraft

text: Aviation played a role in the Brazilian Constitutionalist Revolution of 1932, although both sides had few aircraft. The federal government had approximately 58 aircraft divided between the Navy and the Army, as the Air Force at this time did not constitute an independent branch. In contrast, the rebels had only two Potez 25 planes and two Waco CSO, plus a small number of private aircraft.


title: Attack aircraft

text: During the 1930s, Nazi Germany had begun to field a class of "Schlacht" ("battle") aircraft, such as the Henschel Hs 123. Moreover, the experiences of German Condor Legion during the Spanish Civil War, against an enemy with few fighter aircraft, changed ideas about ground attack. Though equipped with generally unsuitable designs such as the Henschel Hs 123 and cannon-armed versions of the Heinkel He 112, their armament and pilots proved that aircraft were a very effective weapon, even without bombs. This led to some support within the Luftwaffe for the creation of an aircraft dedicated to this role, resulting in tenders for a new "attack aircraft". This led to the introduction (in 1942) of a unique single-seat, twin-engine attack aircraft, the slow-moving but heavily armored and formidably armed Henschel Hs 129 "Panzerknacker" ("Safecracker" /"Tank Cracker").


title: Attack aircraft

text: In Japan, the Imperial Japanese Navy had developed the Aichi D3A dive bomber (based on the Heinkel He 70) and the Mitsubishi B5M light attack bomber. Both, like their US counterparts, were lightly armored types, and were critically reliant on surprise attacks and the absence of significant fighter or AA opposition.


title: Attack aircraft

text: During the Winter War, the Soviet Air Forces used the Polikarpov R-5SSS, and Polikarpov R-ZSh, as attack aircraft.


title: Attack aircraft

text: Perhaps the most notable attack type to emerge during the late 1930s was the Soviet Ilyushin Il-2 "Sturmovik", which became the most-produced military aircraft type in history.


title: Attack aircraft

text: As World War II approached, the concept of an attack aircraft was not well defined, and various air services used many different names for widely differing types, all performing similar roles (sometimes in tandem with non-attack roles of bombers, fighters, reconnaissance and other roles.


title: Attack aircraft

text: The British concept of a light aircraft mixing all the roles that required extensive communication with land forces: reconnaissance, liaison, artillery spotting, aerial supply, and, last but not least, occasional strikes on the battlefield. The concept was similar to front-line aircraft used in the World War I, which was called the CL class in the German Empire. Eventually the RAF's experience showed types such as Westland Lysander to be unacceptably vulnerable and it was replaced by faster fighter types for photoreconnaissance, and light aircraft for artillery spotting.


title: Attack aircraft

text: During the inter-war period, the British considered that in a future war it would be France that would be the enemy. For the light day bomber they had the Fairey Battle which originated in a 1932 specification. Designs in 1938 for a replacement were adapted as a target tug. The last British specification issued for a light bomber was B.20/40 described as a "Close Army Support Bomber" capable of dive bombing and photoreconnaissance. However, the specification was dropped before an aircraft went into production.


title: Attack aircraft

text: In some air services, dive bombers did not equip ground-attack units, but were treated as a separate class. In Nazi Germany, the Luftwaffe distinguished between the "Stuka" ("Sturzkampf-", "dive bombing") units, equipped with Junkers Ju 87 from "Schlacht" ("battle") units, using strafing/low-level bombing types such as the Henschel Hs 123).


title: Attack aircraft

text: Although not a synonymous class with ground-attack aircraft, fighter-bombers were usually used for the role, and proved to excel at it, even when they were only lightly armored. The Royal Air Force and United States Army Air Forces relegated obsolescent fighters to this role, while cutting-edge fighters would serve as interceptors and establish air superiority.


title: Attack aircraft

text: The United States Navy, in distinction to the USAAF, preferred the older term "Scout-Bomber", under a "SB-" designation, such as the Curtiss SB2C Helldiver.


title: Attack aircraft

text: The Junkers Ju 87s of the German Luftwaffe became virtually synonymous with close air support during the early months of World War II. The British Commonwealth's Desert Air Force, led by Arthur Tedder, became the first Allied tactical formation to emphasize the attack role, usually in the form of single-engine Hawker Hurricane and Curtiss P-40 fighter-bombers or specialized "tank-busters", such as the Hurricane Mk IID, armed with two 40 mm Vickers S guns (notably No. 6 Squadron RAF).


title: Attack aircraft

text: At around the same time, a massive invasion by Axis forces had forced the Soviet air forces to quickly expand their army support capacity, such as the Ilyushin Il-2 Sturmovik. The women pilots known as the "Night Witches" utilised an obsolescent, wooden light trainer biplane type, the Polikarpov Po-2 and small anti-personnel bombs in "harassment bombing" attacks that proved difficult to counter.


title: Attack aircraft

text: Wartime experience showed that poorly armored and/or lightly built, pre-war types were unacceptably vulnerable, especially to fighters. Nevertheless, skilled crews could be highly successful in those types, such as the leading "Stuka" ace, Hans-Ulrich Rudel, who claimed 500 tanks, a battleship, a cruiser, and two destroyers in 2,300 combat missions.


title: Attack aircraft

text: The Bristol Beaufighter, based on an obsolescent RAF bomber, became a versatile twin-engine attack aircraft and served in almost every theatre of the war, in the maritime strike and ground attack roles as well as that of night fighter.


title: Attack aircraft

text: Conversely, some mid-war attack types emerged as adaptations of fighters, including several versions of the German Focke-Wulf Fw 190, the British Hawker Typhoon and the US Republic P-47 Thunderbolt. The Typhoon, which was disappointing as a fighter, due to poor high altitude performance, was very fast at low altitudes and thus became the RAF's premier ground attack fighter. It was armed with four 20mm cannon, augmented first with bombs, then rockets. Likewise the P-47 was designed and intended for use as a high altitude bomber escort, but gradually found that role filled by the North American P-51 Mustang (because of its much longer range and greater maneuverability). The P-47 was also heavier and more robust than the P-51 and regarded therefore, as an "energy fighter": ideal for high-speed dive-and-climb tactics, including strafing attacks. Its armament of eight 0.50 caliber machine guns was effective against Axis infantry and light vehicles in both Europe and the Pacific.