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191-493: In particular, the catuṣkoṭi is a "four-cornered" system of argumentation that involves the systematic examination of each of the 4 possibilities of a proposition, P : These four statements hold the following properties: (1) each alternative is mutually exclusive (that is, one of, but no more than one of, the four statements is true) and (2) that all the alternatives are together exhaustive (that is, at least one of them must necessarily be true). This system of logic not only provides

382-453: A diameter in sum. These are the 'eight limits' (Wylie: mtha' brgyad; Sanskrit: aṣṭānta) of 'openness' (Sanskrit: śūnyatā), where śūnyatā is amplified by 'freedom from constructs' or 'simplicity' (Wylie: spros bral; Sanskrit: aprapañca ). Karmay (1988: p. 118) conveys that 'spros bral' is a homologue of 'thig le' (Sanskrit: bindu ), where 'spros bral' is literally "without amplification", understood as "that which cannot be displayed". Thus,

573-452: A sense of urgency (Pali: saṃvega ), deeply moved by the events and their bearing: "Terrible was the quaking, men's hair stood on end, / When the all-accomplished Buddha passed away." Shortly after the council, Ānanda brought the message with regard to the Buddha's directive to Channa personally. Channa was humbled and changed his ways, attained enlightenment, and the penalty was withdrawn by

764-440: A sphere , a mandala , a multidimensional shunya or zero where shunyata denotes zero-ness. The eight arrays are in a concordant relationship where they each constitute a chord to the sphere. The coordinates are equidistant from the epicentre of shunya where the array of the positive configuration (or hemisphere) and the array of the negative configuration (or hemisphere) constitute two polar radii or diametrical complements,

955-492: A stūpa in Ānanda's honor during celebrations and observance days . On a similar note, in 5th–6th-century China and 10th-century Japan, Buddhist texts were composed recommending women to uphold the semi-monastic eight precepts in honor and gratitude of Ānanda. In Japan, this was done through the format of a penance ritual called keka ( Chinese : 悔過 ). By the 13th century, in Japan a cult-like interest for Ānanda had developed in

1146-466: A "theory of eel-wrigglers" in the famous Samannaphala Sutta (DN 2). Sanjaya is recorded as saying: The Catuṣkoṭi, following Nagarjuna, has had a profound impact upon the development of Buddhist logic and its dialectical refinement of Tibetan Buddhism. Robinson (1957: p. 294) qualifies the import of Nagarjuna's work (which includes Nagarjuna's application of the Catuskoti) due to the embedded noise in

1337-488: A Buddha in the future. He would accomplish this slower than the present Buddha, Gotama Buddha , had accomplished this, because Ānanda aspired to becoming a Buddha by applying "great learning". Because of this long trajectory and great efforts, however, his enlightenment would be extraordinary and with great splendor. Ānanda is depicted as an eloquent speaker, who often taught about the self and about meditation. There are numerous Buddhist texts attributed to Ānanda, including

1528-399: A black magic spell to enchant him. This succeeded, and Ānanda was lured into her house, but came to his senses and called upon the help of the Buddha. The Buddha then taught Prakṛti to reflect on the repulsive qualities of the human body, and eventually Prakṛti was ordained as a bhikkhunī , giving up her attachment for Ānanda. In an East Asian version of the story in the Śūraṃgamasūtra ,

1719-530: A certain Daśabāla Kāśyapa. According to the Pāli tradition, Ānanda's first teachers were Belaṭṭhasīsa and Puṇṇa Mantānīputta . It was Puṇṇa's teaching that led Ānanda to attain the stage of sotāpanna (Sanskrit: śrotāpanna ), an attainment preceding that of enlightenment. Ānanda later expressed his debt to Puṇṇa. Another important figure in the life of Ānanda was Sāriputta (Sanskrit: Śāriputra ), one of

1910-415: A crucial role in this council, and texts claim he remembered 84,000 teaching topics, among which 82,000 taught by the Buddha and another 2,000 taught by disciples. Many early Buddhist discourses started with the words " Thus have I heard " (Pali: Evaṃ me sutaṃ , Sanskrit: Evaṃ mayā śrutam ), which according to most Buddhist traditions, were Ānanda's words, indicating that he, as the person reporting

2101-681: A didactic negation. By setting up a series of self-contradictory oppositions, Nāgārjuna disproves all conceivable statements, which can be reduced to these four: With the aid of these four alternatives ( catuṣkoṭika : affirmation, negation, double affirmation, double negation), Nāgārjuna rejects all firm standpoints and traces a middle path between being and nonbeing. Most likely the eight negations, arranged in couplets in Chinese, can be traced back to Nāgārjuna: neither destruction nor production, neither annihilation nor permanence, neither unity nor difference, neither coming nor going. A Mantrayana enumeration of

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2292-771: A distinction with the Greek tradition of the Tetralemma. Also, predicate logic has been applied to the Dharmic Tradition, and though this in some quarters has established interesting correlates and extension of the logico-mathematical traditions of the Greeks, it has also obscured the logico-grammatical traditions of the Dharmic Traditions of Catuṣkoṭi within modern English discourse. The 'Four Extremes' ( Tibetan : མཐའ་བཞི , Wylie : mtha' bzhi ; Sanskrit: Caturanta; Devanagari: चतुरन्त)

2483-488: A drunk and wild elephant released in the Buddha's presence. Ānanda stepped in front of the Buddha to protect him. When the Buddha told him to move, he refused, although normally he always obeyed the Buddha. Through a supernatural accomplishment (Pali: iddhi ; Sanskrit: ṛiddhi ) the Buddha then moved Ānanda aside and subdued the elephant, by touching it and speaking to it with loving-kindness . Ānanda often acted as an intermediary and secretary, passing on messages from

2674-429: A fellow monk Udāyī (Sanskrit: Udāyin ) ridiculed Ānanda. However, the Buddha reprimanded Udāyī in response, saying that Ānanda would certainly be enlightened in this life. The Pāli Mahā-parinibbāna Sutta related the last year-long trip the Buddha took with Ānanda from Rājagaha (Sanskrit: Rājagṛha ) to the small town of Kusinārā (Sanskrit: Kuśinagara ) before the Buddha died there. Before reaching Kusinārā,

2865-462: A final chance to allay any doubts. When no-one responded, Ānanda expressed joy that all of the Buddha's disciples present had attained a level beyond doubts about the Buddha's teaching. However, the Buddha pointed out that Ānanda spoke out of faith and not out of meditative insight —a final reproach. The Buddha added that, of all the five hundred monks that are surrounding him now, even the "latest" or "most backward" (Pali: pacchimaka ) had attained

3056-491: A four-fold configuration resurfaces, with positive and negative configurations being mere rewritten alternatives (ie. it makes no difference whether one works with the positive or negative catuskoti). For example, if one replaces P with Not P, then the positive configuration set for Not P will be functionally equivalent to the negative configuration of P. Sanjaya Belatthiputta , a 6th century BCE Indian ascetic whose teachings are similar to that of Nagarjuna are compared to that of

3247-520: A lot to say about 'dialectic,' but practically nothing to say about formal logic." Robinson (1957: p. 294) opines that Nakamura (1954), developed Schayer's methodology and defended and progressed its application. Robinson (1957: p. 293) opines that the 'metaphysical approach' evident foremost in Murti (1955) was not founded in a firm understanding of the 'logical structure of the system', i.e. catuskoti, for example: Several fundamental limitations of

3438-407: A monk , which was reason enough for Sāṇavāsī to make the decision to get ordained. Though no Early Buddhist Text provides a date for Ānanda's death, according to the Chinese pilgrim monk Faxian (337–422 CE), Ānanda went on to live 120 years. Following the later timeline, however, Ānanda may have lived to 75–85 years. Buddhist studies scholar L. S. Cousins dated Ānanda's death twenty years after

3629-467: A new leader. The final period of the Buddha's life also shows that Ānanda was very much attached to the Buddha's person, and he saw the Buddha's passing with great sorrow. Shortly after the Buddha's death, the First Council was convened, and Ānanda managed to attain enlightenment just before the council started, which was a requirement. He had a historical role during the council as the living memory of

3820-502: A novel method of classifying propositions into logical alternatives, but also because it does so in such a manner that the alternatives are not dependent on the number of truth-values assumed in the system of logic. An example of a Catuṣkoṭi using the arbitrary proposition, "Animals understand love" as P would be: Statements similar to the tetra lemma seems to appear in the Nasadiya Sukta (famous creation hymn), trying to inquire on

4011-553: A number of convents, in which images and stūpas were used and ceremonies were held in his honor. Presently, opinion among scholars is divided as to whether Ānanda's cult among bhikkhunīs was an expression of their dependence on male monastic tradition, or the opposite, an expression of their legitimacy and independence. Pāli Vinaya texts attribute the design of the Buddhist monk's robe to Ānanda. As Buddhism prospered, more laypeople started to donate expensive cloth for robes, which put

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4202-422: A point on the sphere is also called a radius. 'Radius' is used in two senses: as a line segment and also as its length. If a radius is extended through the center to the opposite side of the sphere, it creates a diameter . Like the radius, the length of a diameter is also called the diameter, and denoted d . Diameters are the longest line segments that can be drawn between two points on the sphere: their length

4393-504: A refuge instead. Furthermore, the saṅgha did not have the Buddha as a master anymore, but they would honor those monks who were virtuous and trustworthy. Besides these suttas , a section of the Theragāthā is attributed to Ānanda. Even in the texts attributed to the Buddha himself, Ānanda is sometimes depicted giving a name to a particular text, or suggesting a simile to the Buddha to use in his teachings. In East Asian Buddhism, Ānanda

4584-461: A right-angled triangle connects x , y and r to the origin; hence, applying the Pythagorean theorem yields: Using this substitution gives which can be evaluated to give the result An alternative formula is found using spherical coordinates , with volume element so For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of

4775-489: A secretary and a mouthpiece. Scholars are skeptical about the historicity of many events in Ānanda's life, especially the First Council, and consensus about this has yet to be established. A traditional account can be drawn from early texts, commentaries , and post-canonical chronicles . Ānanda had an important role in establishing the order of bhikkhunīs (Sanskrit: भिक्षुणी , romanized:  bhikṣuṇī , lit.   'female mendicant'), when he requested

4966-605: A spell works? It takes us unawares. Wayman (1977) proffers that the Catuṣkoṭi may be employed in different ways and often these are not clearly stated in discussion nor the tradition. Wayman (1977) holds that the Catuṣkoṭi may be applied in suite, that is all are applicable to a given topic forming a paradoxical matrix; or they may be applied like trains running on tracks (or employing another metaphor, four mercury switches where only certain functions or switches are employed at particular times). This difference in particular establishes

5157-403: A sphere in geography , and the celestial sphere is an important concept in astronomy . Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings . As mentioned earlier r is the sphere's radius; any line from the center to

5348-473: A sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil. In their book Geometry and the Imagination , David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine

5539-512: A sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between " circle " and " disk " in the plane is similar. Small spheres or balls are sometimes called spherules (e.g., in Martian spherules ). In analytic geometry , a sphere with center ( x 0 , y 0 , z 0 ) and radius r

5730-413: A sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space . They draw a distinction between a sphere and a ball , which is a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and

5921-463: A symbol of the Buddha's enlightenment, to give people the chance to pay their respects to the Buddha. This tree and shrine came to be known as the Ānanda Bodhi Tree , said to have grown from a seed from the original Bodhi Tree under which the Buddha is depicted to have attained enlightenment. Many of this type of Bodhi Tree shrines in Southeast Asia were erected following this example. Presently,

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6112-596: A teacher with a "closed fist" would. He also impressed upon Ānanda that he did not think the saṅgha should be reliant too much on a leader, not even himself. He then continued with the well-known statement to take his teaching as a refuge, and oneself as a refuge, without relying on any other refuge, also after he would be gone. Bareau argued that this is one of the most ancient parts of the text, found in slight variation in five early textual traditions: "Moreover, this very beautiful episode, touching with nobility and psychological verisimilitude with regard to both Ānanda and

6303-401: A total of 25 years, a duty which entailed much work. His relationship with the Buddha is depicted as warm and trusting: when the Buddha grew ill, Ānanda had a sympathetic illness; when the Buddha grew older, Ānanda kept taking care of him with devotion. Ānanda sometimes literally risked his life for his teacher. At one time, the rebellious monk Devadatta tried to kill the Buddha by having

6494-407: A unique circle in a plane. Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle. By examining the common solutions of the equations of two spheres , it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. Although the radical plane

6685-621: A view of ' svabhava ': Svabhava is by defini[t]ion the subject of contradictory ascriptions. If it exists, it must belong to an existent entity, which means that it must be conditioned, dependent on other entities, and possessed of causes. But a svabhava is by definition unconditioned, not dependent on other entities, and not caused. Thus the existence of a svabhava is impossible. [ NB: typographical errors repaired ] "Nature" (a gloss of prakrti which in this context equals svabhava) does not entail an alter-entity: The term "nature" (prakrti equals svabhava) has no complement..."If (anythings) existence

6876-516: Is paritta recitation. Theravāda Buddhists explain that the custom of sprinkling water during paritta chanting originates in Ānanda's visit to Vesālī, when he recited the Ratana Sutta and sprinkled water from his alms bowl. A third tradition sometimes attributed to Ānanda is the use of Bodhi trees in Buddhism. It is described in the text Kāliṅgabodhi Jātaka that Ānanda planted a Bodhi tree as

7067-659: Is a particular application of the Catuṣkoṭi: Dumoulin et al.' s (1988, 2005: pp. 43–44), work on Zen models a particular formulation of the Catuṣkoṭi that approaches the Caturanta engaging the Buddhist notion of dharmas , and attribute the model to Nagarjuna : If we focus on the doctrinal agreement that exists between the Wisdom Sūtras and the tracts of the Mādhyamika we note that both schools characteristically practice

7258-462: Is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point). The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection. They intersect at right angles (are orthogonal ) if and only if

7449-433: Is an equation of a sphere whose center is P 0 {\displaystyle P_{0}} and whose radius is ρ {\displaystyle {\sqrt {\rho }}} . If a in the above equation is zero then f ( x , y , z ) = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity . A parametric equation for

7640-634: Is another world. But I don't say so. And I don't say otherwise. And I don't say it is not, and I don't not say it is not." "Is there no other world?..." "Is there both another world and no other world?..."Is there neither another world nor no other world?..." "Are there spontaneously-born beings?..." "Are there not...?" "Both...? "Neither...?" "Does the Tathagata exist after death? Does he not exist after death? Does he both exist and not exist after death? Does he neither exist nor not exist after death?..." "If I thought so, I would say so...I don't say so...I don't say it

7831-485: Is considered one of the ten principal disciples . In many Indian Sanskrit and East Asian texts, Ānanda is considered the second patriarch of the lineage which transmitted the teaching of the Buddha, with Mahākassapa being the first and Majjhantika or Saṇavāsī being the third. There is an account dating back from the Sarvāstivāda and Mūlasarvāstivāda textual traditions which states that before Mahākassapa died, he bestowed

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8022-471: Is due to its nature, its non-existence will not occur, since the alter-entity (complement) of a nature never occurs." ( Mūlamadhyamakakārikā , 15.8) That is, a nature is the class of properties attributed to a class of terms. Since they are necessarily present throughout the range of the subject or class of subjects, cases of their absence do not occur. Y Karunadasa (1999, 2000: p. 1) holds that Early Buddhism and early Buddhist discourse "often refer to

8213-651: Is historically certain, Cousins thought it likely at least one of the leading figures on the Second Council was a pupil of Ānanda, as nearly all the textual traditions mention a connection with Ānanda. Ajāsattu is said to have built a stūpa on top of the Ānanda's relics, at the river Rohīni, or according to some sources, the Ganges; the Licchavis had also built a stūpa at their side of the river. The Chinese pilgrim Xuan Zang (602–64 CE) later visited stūpas on both sides of

8404-487: Is identical to the Catuṣkoṭi (and there are numerous other analogues of this fourfold grammatical structure within this Sutta), the intentionality of the architecture employed by Nagarjuna is not evident, as rendered into English by Walshe (1987, 1995: p. 81): 'What is the fourth way? Here, an ascetic or Brahmin is dull and stupid. Because of his dullness and stupidity, when he is questioned he resorts to evasive statements and wriggles like an eel: "If you ask me whether there

8595-546: Is known as the Treasurer of the Dhamma , with Dhamma ( Sanskrit : धर्म , dharma ) referring to the Buddha's teaching. In Early Buddhist Texts , Ānanda was the first cousin of the Buddha. Although the early texts do not agree on many parts of Ānanda's early life, they do agree that Ānanda was ordained as a monk and that Puṇṇa Mantānīputta ( Sanskrit : पूर्ण मैत्रायणीपुत्र , Pūrṇa Maitrāyaṇīputra) became his teacher. Twenty years in

8786-468: Is mentioned more often than any other disciple: he is named foremost in conduct, in attention to others, in power of memory, in erudition and in resoluteness. Ānanda was the subject of a sermon of praise delivered by the Buddha just before the Buddha's death, as described in the Mahāparinibbāna Sutta : it is a sermon about a man who is kindly, unselfish, popular, and thoughtful toward others. In

8977-452: Is not preceded by any account of Ānanda learning Abhidhamma. According to some later Mahāyāna accounts, Ānanda also assisted in reciting Mahāyāna texts, held in a different place in Rājagaha, but in the same time period. The Pāli commentaries state that after the council, when the tasks for recitation and memorizing the texts were divided, Ānanda and his pupils were given the task to remember

9168-471: Is not." This is the fourth case.' McEvilley (2002: p. 495) makes a case for mutual iteration and pervasion between Pyrrhonism and Madhyamika : An extraordinary similarity, that has long been noticed, between Pyrrhonism and Mādhyamika is the formula known in connection with Buddhism as the fourfold negation ( catuṣkoṭi ) and which in Pyrrhonic form might be called the fourfold indeterminacy. The Catuṣkoṭi

9359-405: Is principally built upon, though divergent from, the work of Nakamura: In considering the formal structure of Nagarjuna's argumentation, I exclude epistemology, psychology, and ontology from consideration.... Such extra-logical observations as emerge will be confined to the concluding paragraphs... Puhakka (2003: p. 134-145) charts the stylized reification process of a human sentient being ,

9550-413: Is the center of the sphere, and r is the sphere's radius . The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians . The sphere is a fundamental object in many fields of mathematics . Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as

9741-497: Is the locus of all points ( x , y , z ) such that Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface , a type of algebraic surface . Let a, b, c, d, e be real numbers with a ≠ 0 and put Then the equation has no real points as solutions if ρ < 0 {\displaystyle \rho <0} and is called the equation of an imaginary sphere . If ρ = 0 {\displaystyle \rho =0} ,

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9932-510: Is the irrationality of Existence, or the falsehood of reasoning which is built upon the logical principle that A equals A.... Because two answers, assertion and denial, are always possible to a given question, his arguments contain two refutations, one denying the presence, one the absence of the probandum. This double refutation is called the Middle Path. [ emphasis evident in Robinson ] Śūnyatā

10123-426: Is the ninth ' view ' (Sanskrit: dṛṣṭi), the viewless view, a superposition of the eight possible arrays of proposition P [and its 'inseparable contradistinction' (Sanskrit: apoha )]. The eight arrays or octaves of the iconographic Dharmacakra represent drishti or traditional views that Shakyamuni countered. These eight arrays may be plotted as coordinates on a multidimensional field which may be rendered as

10314-511: Is the summation of all shell volumes: In the limit as δr approaches zero this equation becomes: Substitute V : Differentiating both sides of this equation with respect to r yields A as a function of r : This is generally abbreviated as: where r is now considered to be the fixed radius of the sphere. Alternatively, the area element on the sphere is given in spherical coordinates by dA = r sin θ dθ dφ . The total area can thus be obtained by integration : The sphere has

10505-525: Is true even in the simple act of perception when we see a figure that we become aware of as "something." As the German gestalt psychologists demonstrated, for each figure perceived, there is a background of which we remain relatively unaware. We can extend this to texts or spoken communications. For every text we understand there is a context we are not fully cognizant of. Thus, with every figure noticed or reality affirmed, there is, inevitably, unawareness. Is this how

10696-401: Is twice the radius, d = 2 r . Two points on the sphere connected by a diameter are antipodal points of each other. A unit sphere is a sphere with unit radius ( r = 1 ). For convenience, spheres are often taken to have their center at the origin of the coordinate system , and spheres in this article have their center at the origin unless a center is mentioned. A great circle on

10887-424: Is usually not depicted in this manner, and the motif of transmission of the Dhamma through a list of patriarchs is not found in Pāli sources. Because Ānanda was instrumental in founding the bhikkhunī community, he has been honored by bhikkhunīs for this throughout Buddhist history. The earliest traces of this can be found in the writings of Faxian and Xuan Zang, who reported that bhikkhunīs made offerings to

11078-501: Is within the framework of the Buddhist critique of sassatavada and ucchadavada that the Buddhist doctrines seem to assume their significance. For it is through the demolition of these two world-views that Buddhism seeks to construct its own world-view. The conclusion is that it was as a critical response to the mutual opposition between these two views that Buddhism emerged as a new faith amidst many other faiths. Ananda Ānanda ( Pali and Sanskrit : आनंद; 5th–4th century BCE)

11269-778: The Atthakanāgara Sutta , about meditation methods to attain Nirvana; a version of the Bhaddekaratta Sutta (Sanskrit: Bhadrakārātrī , pinyin : shanye ), about living in the present moment; the Sekha Sutta , about the higher training of a disciple of the Buddha; the Subha Suttanta , about the practices the Buddha inspired others to follow. In the Gopaka-Mogallānasutta , a conversation took place between Ānanda,

11460-537: The Divyavadāna and the Aśokavadāna , before the Buddha's death, the Buddha confided to Ānanda that the latter's student Majjhantika (Sanskrit: Madhyāntika ) would travel to Udyāna, Kashmir , to bring the teaching of the Buddha there. Mahākassapa made a prediction that later would come true that another of Ānanda's future pupils, Sāṇavāsī (Sanskrit: Śāṇakavāsī, Śāṇakavāsin or Śāṇāvasika ), would make many gifts to

11651-478: The Dīgha Nikāya . During the same council, Ānanda was charged for an offense by members of the saṅgha for having enabled women to join the monastic order. Besides this, he was charged for having forgotten to request the Buddha to specify which offenses of monastic discipline could be disregarded; for having stepped on the Buddha's robe; for having allowed women to honor the Buddha's body after his death, which

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11842-449: The Second and Third Councils . Ānanda died 20 years after the Buddha, and stūpas (monuments) were erected at the river where he died. Ānanda is one of the most loved figures in Buddhism. He was known for his memory, erudition and compassion, and was often praised by the Buddha for these matters. He functioned as a foil to the Buddha, however, in that he still had worldly attachments and

12033-459: The Vajji confederacy. According to later texts, an enlightened monk also called Vajjiputta (Sanskrit: Vajraputra ) had an important role in Ānanda's life. He listened to a teaching of Ānanda and realized that Ānanda was not enlightened yet. Vajjiputta encouraged Ānanda to talk less to laypeople and deepen his meditation practice by retreating in the forest, advice that very much affected Ānanda. In

12224-410: The bhikkhunī order may have well been initiated by Ānanda after the Buddha's death, and the introduction of Mahāpajāpati as the person requesting to do so is merely a literary device to connect the ordination of women with the person of the Buddha, through his foster mother. Von Hinüber concludes this based on several patterns in the early texts, including the apparent distance between the Buddha and

12415-553: The bhikkhunī order, and the frequent discussions and differences of opinion that take place between Ānanda and Mahākassapa. Some scholars have seen merits in von Hinüber's argument with regard to the pro- and anti-factions, but as of 2017, no definitive evidence has been found for the theory of establishment of the bhikkhuni order after the Buddha's death. Buddhist studies scholar Bhikkhu Anālayo has responded to most of von Hinuber's arguments, writing: "Besides requiring too many assumptions, this hypothesis conflicts with nearly 'all

12606-410: The sadhana of the ' mūla prajñā ', as definitions are slippery and challenging to pinpoint that hold for all contexts. Language usage in Buddhist logic is not intuitive but technical and must be learnt, acquired through the perfection and power of 'diligence' (Sanskrit: vīrya ). The following quotations are cited to provide insight ( in lieu of technical definitions) into the understanding of

12797-440: The saṅgha at Mathurā , during a feast held from profits of successful business. After this event, Ānanda would successfully persuade Sāṇavāsī to become ordained and be his pupil. Ānanda later persuaded Sāṇavāsī by pointing out that the latter had now made many material gifts, but had not given " the gift of the Dhamma ". When asked for explanation, Ānanda replied that Sāṇavāsī would give the gift of Dhamma by becoming ordained as

12988-466: The saṅgha , and more generally, that now the time had come for the Buddhist monastics and devotees to take the Buddhist texts as authority, now that the Buddha was dying. The Buddha gave several instructions before his death, including a directive that his former charioteer Channa (Sanskrit: Chandaka ) be shunned by his fellow monks, to humble his pride. In his final moments, the Buddha asked if anyone had any questions they wished to pose to him, as

13179-414: The saṅgha . Ānanda traveled to Sāvatthī (Sanskrit: Śrāvastī ), where he was met with a sad populace, who he consoled with teachings on impermanence. After that, Ānanda went to the quarters of the Buddha and went through the motions of the routine he formerly performed when the Buddha was still alive, such as preparing water and cleaning the quarters. He then saluted and talked to the quarters as though

13370-414: The volume inside a sphere (that is, the volume of a ball , but classically referred to as the volume of a sphere) is where r is the radius and d is the diameter of the sphere. Archimedes first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere (having the height and diameter equal to the diameter of

13561-399: The x -axis from x = − r to x = r , assuming the sphere of radius r is centered at the origin. At any given x , the incremental volume ( δV ) equals the product of the cross-sectional area of the disk at x and its thickness ( δx ): The total volume is the summation of all incremental volumes: In the limit as δx approaches zero, this equation becomes: At any given x ,

13752-459: The Buddha allow him to accept invitations on his behalf, allow him to ask questions about his doctrine, and repeat any teaching that the Buddha had taught in Ānanda's absence. These requests would help people trust Ānanda and show that the Buddha was sympathetic to his attendant. Furthermore, Ānanda considered these the real advantages of being an attendant, which is why he requested them. The Buddha agreed to Ānanda's conditions, and Ānanda became

13943-462: The Buddha and the lay communities. After the Buddha's death, some sources say Ānanda stayed mostly in the West of India, in the area of Kosambī (Sanskrit: Kausambī ), where he taught most of his pupils. Other sources say he stayed in the monastery at Veḷuvana (Sanskrit: Veṇuvana ). Several pupils of Ānanda became well-known in their own right. According to post-canonical Sanskrit sources such as

14134-746: The Buddha as a monk, or had his brother do so. Ānanda used this opportunity, and asked his brother Devadatta to stay at home, so that he could leave for the monkhood. The later timeline from the Mūlasarvāstivāda texts and the Pāli Theragāthā , however, have Ānanda ordain much later, about twenty-five years before the Buddha's death —in other words, twenty years in the Buddha's ministry. Some Sanskrit sources have him ordain even later. The Mūlasarvāstivāda texts on monastic discipline (Pāli and Sanskrit: Vinaya ) relate that soothsayers predicted Ānanda would be

14325-435: The Buddha at the wrong time. For example, one time Mahākassapa chastised Ānanda in strong words, criticizing the fact that Ānanda was travelling with a large following of young monks who appeared untrained and who had built up a bad reputation. In another episode described in a Sarvāstivāda text, Ānanda is the only disciple who was willing to teach psychic powers to Devadatta, who later would use these in an attempt to destroy

14516-455: The Buddha did not accept them. All the while Ānanda remained quiet. When he was asked why, he said that the Buddha would know best whom to choose, upon which the Buddha responded by choosing Ānanda. Ānanda agreed to take on the position, on the condition that he did not receive any material benefits from the Buddha. Accepting such benefits would open him up to criticism that he chose the position because of ulterior motives. He also requested that

14707-452: The Buddha intended all along to establish a bhikkhunī order. Regardless, during the acceptance of women into the monastic order, the Buddha told Ānanda that the Buddha's Dispensation would last shorter because of this. At the time, the Buddhist monastic order consisted of wandering celibate males, without many monastic institutions. Allowing women to join the Buddhist celibate life might have led to dissension, as well as temptation between

14898-474: The Buddha on behalf of the latter's foster-mother Mahāpajāpati Gotamī ( Sanskrit : महाप्रजापती गौतमी , Mahāprajāpatī Gautamī ) to allow her to be ordained. Ānanda also accompanied the Buddha in the last year of his life, and therefore was witness to many tenets and principles that the Buddha conveyed before his death, including the well-known principle that the Buddhist community should take his teaching and discipline as their refuge, and that he would not appoint

15089-482: The Buddha prohibited Ānanda from being involved himself, but rather had him instruct the Mallas to perform the rituals, these instructions have by many scholars been interpreted as a prohibition that monastics should not be involved in funerals or worship of stūpas (structures with relics). Buddhist studies scholar Gregory Schopen has pointed out, however, that this prohibition only held for Ānanda, and only with regard to

15280-417: The Buddha sent Mañjuśrī to help Ānanda, who used recitation to counter the magic charm. The Buddha then continued by teaching Ānanda and other listeners about the Buddha nature . In the role of mediator between the Buddha and the lay communities, Ānanda sometimes made suggestions to the Buddha for amendments in the monastic discipline. Most importantly, the early texts attribute the inclusion of women in

15471-670: The Buddha spent the retreat during the monsoon (Pali: vassa , Sanskrit: varṣā ) in Veḷugāma (Sanskrit: Veṇugrāmaka ), getting out of the Vesālī area which suffered from famine. Here, the eighty-year old Buddha expressed his wish to speak to the saṅgha once more. The Buddha had grown seriously ill in Vesālī, much to the concern of some of his disciples. Ānanda understood that the Buddha wished to leave final instructions before his death. The Buddha stated, however, that he had already taught everything needed, without withholding anything secret as

15662-401: The Buddha taught Ānanda is the passage about spiritual friendship (Pali: kalyāṇamittata ). In this passage, Ānanda stated that spiritual friendship is half of the holy life; the Buddha corrected Ānanda, stating that such friendship is the entire holy life. In summary, Ānanda worked as an assistant, intermediary and a mouthpiece, helping the Buddha in many ways, and learning his teachings in

15853-506: The Buddha was still there. The Pāli commentaries state that Ānanda did this out of devotion, but also because he was "not yet free from the passions ". According to the texts, the First Buddhist Council was held in Rājagaha. In the first vassa after the Buddha had died, the presiding monk Mahākassapa (Sanskrit: Mahākāśyapa ) called upon Ānanda to recite the discourses he had heard, as a representative on this council. There

16044-480: The Buddha whether women could become enlightened as well. Although the Buddha conceded this, he did not allow the Sākiya women to be ordained yet. Ānanda then discussed with the Buddha how Mahāpajāpatī took care of him during his childhood, after the death of his real mother . Ānanda also mentioned that previous Buddhas had also ordained bhikkhunīs . In the end, the Buddha allowed the Sākiya women to be ordained, being

16235-400: The Buddha's attendant, accompanying the Buddha on most of his wanderings. Ānanda took care of the Buddha's daily practical needs, by doing things such as bringing water and cleaning the Buddha's dwelling place . He is depicted as observant and devoted, even guarding the dwelling place at night. Ānanda takes the part of interlocutor in many of the recorded dialogues. He tended the Buddha for

16426-472: The Buddha's attendant. In order to prevent Ānanda from leaving the palace to ordain, his father brought him to Vesālī (Sanskrit: Vaiśālī ) during the Buddha's visit to Kapilavatthu, but later the Buddha met and taught Ānanda nonetheless. On a similar note, the Mahāvastu relates, however, that Mṛgī was initially opposed to Ānanda joining the holy life, because his brother Devadatta had already ordained and left

16617-423: The Buddha's conditions. Prakṛti agrees, and it is revealed that the Buddha means something else than she does: he asks Prakṛti to ordain as a bhikkhunī , and live the celibate life as a kind of sister to Ānanda. At first, Prakṛti weeps in dismay, but after the Buddha explains that her current situation is a result of karma from her previous life, she understands and rejoices in the life of a bhikkhunī . Apart from

16808-438: The Buddha's enlightenment. Furthermore, Mahāpajāpati was the Buddha's foster mother, and must therefore have been considerably older than him. However, after the bhikkhunī order was established, Mahāpajāpati still had many audiences with the Buddha, as reported in Pāli and Chinese Early Buddhist Texts. Because of this and other reasons, it could be inferred that establishment of the bhikkhunī order actually took place early in

16999-457: The Buddha's funeral ceremony. It has also been shown that the instructions on the funeral are quite late in origin, in both composition and insertion into the text, and are not found in parallel texts, apart from the Mahāparinibbāna Sutta . Ānanda then continued by asking how devotees should honor the Buddha after his death. The Buddha responded by listing four important places in his life that people could pay their respects to, which later became

17190-459: The Buddha's initial refusal rather as a test of resolve, following a widespread pattern in the Pāli Canon and in monastic procedure of repeating a request three times before final acceptance. Some also argue that the Buddha was believed by Buddhists to be omniscient , and therefore is unlikely to have been depicted as changing his mind. Other scholars argue that other passages in the texts indicate

17381-523: The Buddha's main disciples. Sāriputta often taught Ānanda about the finer points of Buddhist doctrine; they were in the habit of sharing things with one another, and their relationship is described as a good friendship. In some Mūlasarvāstivāda texts, an attendant of Ānanda is also mentioned who helped motivate Ānanda when he was banned from the First Buddhist Council . He was a "Vajjiputta" (Sanskrit: Vṛjjiputra ), i.e. someone who originated from

17572-427: The Buddha's message on to him. When Ānanda was crossing the river, he was followed by King Ajātasattu (Sanskrit: Ajātaśatrū ), who wanted to witness his death and was interested in his remains as relics. Ānanda had once promised Ajāsattu that he would let him know when he would die, and accordingly, Ānanda had informed him. On the other side of the river, however, a group of Licchavis from Vesālī awaited him for

17763-457: The Buddha's ministry, Ānanda became the attendant of the Buddha, when the Buddha selected him for this task. Ānanda performed his duties with great devotion and care, and acted as an intermediary between the Buddha and the laypeople, as well as the saṅgha ( Sanskrit : संघ , romanized :  saṃgha , lit.   'monastic community'). He accompanied the Buddha for the rest of his life, acting not only as an assistant, but also

17954-474: The Buddha's ministry. If this is the case, Ānanda's role in establishing the order becomes less likely. Some scholars therefore interpret the names in the account, such as Ānanda and Mahāpajāpati , as symbols, representing groups rather than specific individuals. According to the texts, Ānanda's role in founding the bhikkhunī order made him popular with the bhikkhunī community. Ānanda often taught bhikkhunīs , often encouraged women to ordain, and when he

18145-407: The Buddha's teaching on Ānanda as a formal passing on of authority, telling Ānanda to pass the teaching on to Ānanda's pupil Saṇavāsī. Later, just before Ānanda died, he did as Mahākassapa had told him to. Buddhist studies scholars Akira Hirakawa and Bibhuti Baruah have expressed skepticism about the teacher–student relationship between Mahākassapa and Ānanda, arguing that there was discord between

18336-526: The Buddha's teachings accurately and without distortion, was "both a gift and a burden". Ānanda was able to remember many discourses verbatim, but this also went hand-in-hand with a habit of not reflecting on those teachings, being afraid that reflection might distort the teachings as he heard them. At multiple occasions, Ānanda was warned by other disciples that he should spend less time on conversing to lay people, and more time on his own practice. Even though Ānanda regularly practiced meditation for long hours, he

18527-462: The Buddha's teachings. He also taught other disciples to memorize Buddhist doctrine. For these reasons, Ānanda became known as the "Treasurer of the Dhamma " (Pali: Dhamma-bhaṇḍāgārika , Sanskrit: Dharma-bhaṇḍāgārika ), Dhamma (Sanskrit: Dharma ) referring to the doctrine of the Buddha. Being the person who had accompanied the Buddha throughout a great part of his life, Ānanda was in many ways

18718-431: The Buddha's. Ānanda was teaching till the end of his life. According to Mūlasarvāstivāda sources, Ānanda heard a young monk recite a verse incorrectly, and advised him. When the monk reported this to his teacher, the latter objected that "Ānanda has grown old and his memory is impaired ..." This prompted Ānanda to attain final Nirvana. He passed on the "custody of the [Buddha's] doctrine" to his pupil Sāṇavāsī and left for

18909-458: The Buddha, informing the Buddha of news, invitations, or the needs of lay people, and advising lay people who wanted to provide gifts to the saṅgha . At one time, Mahāpajāpatī , the Buddha's foster-mother, requested to offer robes for personal use for the Buddha. She said that even though she had raised the Buddha in his youth, she never gave anything in person to the young prince; she now wished to do so. The Buddha initially insisted that she give

19100-591: The Buddha, reciting many of the Buddha's discourses and checking them for accuracy. During the same council, however, he was chastised by Mahākassapa ( Sanskrit : महाकाश्यप , Mahākāśyapa ) and the rest of the saṅgha for allowing women to be ordained and failing to understand or respect the Buddha at several crucial moments. Ānanda continued to teach until the end of his life, passing on his spiritual heritage to his pupils Sāṇavāsī ( Sanskrit : शाणकवासी , Śāṇakavāsī ) and Majjhantika ( Sanskrit : मध्यान्तिक , Madhyāntika ), among others, who later assumed leading roles in

19291-619: The Buddha, seems to go back very far, at the time when the authors, like the other disciples, still considered the Blessed One [the Buddha] a man, an eminently respectable and undefiled master, to whom behavior and utterly human words were lent, so that one is even tempted to see there the memory of a real scene which Ānanda reportedly told to the Community in the months following the Parinirvāṇa [death of

19482-452: The Buddha. According to a Mahīśāsaka text, however, when Devadatta had turned against the Buddha, Ānanda was not persuaded by him, and voted against him in a formal meeting. Ānanda's late spiritual growth is much discussed in Buddhist texts, and the general conclusion is that Ānanda was slower than other disciples due to his worldly attachments and his attachment to the person of the Buddha, both of which were rooted in his mediating work between

19673-480: The Buddha. At night, staff and torch in hand, he went nine times round the Buddha's cell and never put them down lest he would fall asleep and fail to answer a call to the Buddha." Ānanda was recognized as one of the most important disciples of the Buddha. In the lists of the disciples given in the Aṅguttara Nikāya and Saṃyutta Nikāya , each of the disciples is declared to be foremost in some quality. Ānanda

19864-413: The Buddha. Mahākassapa then listed seven charges to counter Ānanda's objection. The charges were similar to the five given in Pāli. Other textual traditions list slightly different charges, amounting to a combined total of eleven charges, some of which are only mentioned in one or two textual traditions. Considering that an enlightened disciple was seen to have overcome all faults, it seems more likely that

20055-442: The Buddha]." The same text contains an account in which the Buddha, at numerous occasions, gave a hint that he could prolong his life to a full eon through a supernatural accomplishment, but this was a power that he would have to be asked to exercise. Ānanda was distracted, however, and did not take the hint. Later, Ānanda did make the request, but the Buddha replied that it was already too late, as he would die soon. Māra ,

20246-464: The Buddhist personification of evil, had visited the Buddha, and the Buddha had decided to die in three months. When Ānanda heard this, he wept. The Buddha consoled him, however, pointing out that Ānanda had been a great attendant, being sensitive to the needs of different people. If he was earnest in his efforts, he would attain enlightenment soon. He then pointed out to Ānanda that all conditioned things are impermanent : all people must die. In

20437-548: The Catuskoti (Tetralemma) as an attribute of propositional logic and critiques Stcherbatsky. Robinson (1957: p. 294) states that "Schayers criticisms of Stcherbatsky are incisive and just." Murti (1955) makes no mention of the logical contribution of Schayer. According to Robinson (1957: p. 294), Murti furthered the work of Stcherbatsky amongst others, and brought what Robinson terms "the metaphysical phase of investigation" to its apogee though qualifies this with: "Murti has

20628-583: The English rendering following Dowman (2003) and Wylie following Norbu et al. (2001): Within English Buddhist logico-epistemological discourse, there is and has been historically, much obstruction to an understanding of the Caturanta (as the Catuṣkoṭi) due to inherent negligence in terminology not being clearly defined from the outset. That said, acquisition of terminology must be engaged and actualized though

20819-486: The First Council lasted for seven months. Scholars doubt, however, whether the entire canon was really recited during the First Council, because the early texts contain different accounts on important subjects such as meditation. It may be, though, that early versions were recited of what is now known as the Vinaya-piṭaka and Sutta-piṭaka . Nevertheless, many scholars, from the late 19th century onward, have considered

21010-511: The First Council was written after the Second Council , and based on that of the Second, since there were not any major problems to solve after the Buddha's death, or any other need to organize the First Council. Much material in the accounts, and even more so in the more developed later accounts, deal with Ānanda as the unsullied intermediary who passes on the legitimate teaching of the Buddha. On

21201-900: The Four Limits or the Four Extremes within the Buddhadharma is also common. These four limits are evident in the earliest sutras of the Theravadin of the First Turning, through the Second Turning philosophy of Nagarjuna and his disciples and commentators and also evident in the Third Turning as evidenced in the presentation of Padmasambhava. Padmasambhava in his "Secret Instruction in a Garland of Vision" ( Tibetan : མན་ངག་ལྟ་བའི་ཕྲེང་བ , Wylie : man ngag lta ba'i phreng ba ) lists them as follows with

21392-516: The Pali Canon do preserve for us the Buddha-vacana , 'the Buddha's words', transmitted to us via his disciple Ānanda and the First Council". "He served the Buddha following him everywhere like a shadow, bringing him tooth wood and water, washing his feet, rubbing his body, cleaning his cell and fulfilling all his duties with the greatest care. By day he was at hand forestalling the slightest wish of

21583-401: The Pāli tradition, the charges were laid after Ānanda had become enlightened and done all the recitations; but the Mūlasarvāstivāda tradition has it that the charges were laid before Ānanda became enlightened and started the recitations. In this version, when Ānanda heard that he was banned from the council, he objected that he had not done anything that went against the teaching and discipline of

21774-542: The Tathāgata is the body of dharma(s), pure in nature. He [Ānanda] was able to retain it/them all; for this reason the offerings [to him] surpass [all others]"— body of dharma here referred to the Buddha's teachings as a whole. In Early Buddhist Texts, Ānanda had reached final Nirvana and would no longer be reborn. But, in contrast with the early texts, according to the Mahāyāna Lotus Sūtra , Ānanda would be born as

21965-472: The above stated equations as where ρ is the density (the ratio of mass to volume). A sphere can be constructed as the surface formed by rotating a circle one half revolution about any of its diameters ; this is very similar to the traditional definition of a sphere as given in Euclid's Elements . Since a circle is a special type of ellipse , a sphere is a special type of ellipsoid of revolution . Replacing

22156-470: The accounts which lead to the offenses Ānanda was charged with during the First Council. Moreover, Ānanda's weaknesses described in the texts were that he was sometimes slow-witted and lacked mindfulness, which became noticeable because of his role as attendant to the Buddha: this involved minor matters like deportment, but also more important matters, such as ordaining a man with no future as a pupil, or disturbing

22347-512: The brahmin Gopaka-Mogallāna and the minister Vassakara, the latter being the highest official of the Magadha region. During this conversation, which occurred shortly after the Buddha's death, Vassakara asked whether it was decided yet who would succeed the Buddha. Ānanda replied that no such successor had been appointed, but that the Buddhist community took the Buddha's teaching and discipline as

22538-416: The charges were laid before Ānanda's attainment than after. Indologists von Hinüber and Jean Przyluski argue that the account of Ānanda being charged with offenses during the council indicate tensions between competing early Buddhist schools , i.e. schools that emphasized the discourses (Pali: sutta , Sanskrit: sūtra ) and schools that emphasized monastic discipline. These differences have affected

22729-442: The circle with an ellipse rotated about its major axis , the shape becomes a prolate spheroid ; rotated about the minor axis, an oblate spheroid. A sphere is uniquely determined by four points that are not coplanar . More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. This property is analogous to the property that three non-collinear points determine

22920-498: The council started. The Mūlasarvāstivāda texts add that he felt motivated when he remembered the Buddha's words that he should be his own refuge, and when he was consoled and advised by Anuruddha and Vajjiputta, the latter being his attendant. On the night before the event, he tried hard to attain enlightenment. After a while, Ānanda took a break and decided to lie down for a rest. He then attained enlightenment right there, right then, halfway between standing and lying down. Thus, Ānanda

23111-431: The council was required, he did not want to be biased by allowing an exception to the rule. The Mūlasarvāstivāda tradition adds that Mahākassapa initially allowed Ānanda to join as a sort of servant assisting during the council, but then was forced to remove him when the disciple Anuruddha saw that Ānanda was not yet enlightened. Ānanda felt humiliated, but was prompted to focus his efforts to reach enlightenment before

23302-423: The cube, since V = ⁠ π / 6 ⁠ d , where d is the diameter of the sphere and also the length of a side of the cube and ⁠ π / 6 ⁠  ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1   m, or about 0.524 m . The surface area of a sphere of radius r is: Archimedes first derived this formula from

23493-422: The discourses and to determine which were authentic and which were not. Mahākassapa asked of each discourse that Ānanda listed where, when, and to whom it was given, and at the end of this, the assembly agreed that Ānanda's memories and recitations were correct, after which the discourse collection (Pali: Sutta Piṭaka , Sanskrit: Sūtra Piṭaka ) was considered finalized and closed. Ānanda therefore played

23684-401: The discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness. At any given radius r , the incremental volume ( δV ) equals the product of the surface area at radius r ( A ( r ) ) and the thickness of a shell ( δr ): The total volume

23875-410: The draft was used by composer Jonathan Harvey in his 2007 opera Wagner Dream . In Wagner's version of the legend, which he based on orientalist Eugène Burnouf 's translations, the magical spell of Prakṛti's mother does not work on Ānanda, and Prakṛti turns to the Buddha to explain her desires for Ānanda. The Buddha replies that a union between Prakṛti and Ānanda is possible, but Prakṛti must agree to

24066-571: The early saṅgha (monastic order) to Ānanda. Fifteen years after the Buddha's enlightenment, his foster mother Mahāpajāpatī came to see him to ask him to be ordained as the first Buddhist bhikkhunī . Initially, the Buddha refused this. Five years later, Mahāpajāpatī came to request the Buddha again, this time with a following of other Sākiya women, including the Buddha's former wife Yasodharā (Sanskrit: Yaśodarā ). They had walked 500 kilometres (310 miles), looked dirty, tired and depressed, and Ānanda felt pity for them. Ānanda therefore confirmed with

24257-412: The evidence preserved in the texts together'", arguing that it was monastic discipline that created a distance between the Buddha and the bhikkhunīs , and even so, there were many places in the early texts where the Buddha did address bhikkhunīs directly. Despite his long association with and close proximity to the Buddha, the texts describe that Ānanda had not become enlightened yet. Because of that,

24448-523: The fact that the projection to the lateral surface of a circumscribed cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r . At infinitesimal thickness

24639-562: The final days of the Buddha's life, the Buddha traveled to Kusinārā. The Buddha had Ānanda prepare a place for lying down between two sal trees , the same type of tree under which the mother of the Buddha gave birth. The Buddha then had Ānanda invite the Malla clan from Kusinārā to pay their final respects. Having returned, Ānanda asked the Buddha what should be done with his body after his death, and he replied that it should be cremated, giving detailed instructions on how this should be done. Since

24830-407: The first twenty years of the Buddha's ministry, the Buddha had several personal attendants. However, after these twenty years, when the Buddha was aged 55, the Buddha announced that he had need for a permanent attendant. The Buddha had been growing older, and his previous attendants had not done their job very well. Initially, several of the Buddha's foremost disciples responded to his request, but

25021-444: The four main places of Buddhist pilgrimage . Before the Buddha died, Ānanda recommended the Buddha to move to a more meaningful city instead, but the Buddha pointed out that the town was once a great capital. Ānanda then asked who will be next teacher after the Buddha would be gone, but the Buddha replied that his teaching and discipline would be the teacher instead. This meant that decisions should be made by reaching consensus within

25212-410: The historicity of the First Council improbable. Some scholars, such as orientalists Louis de La Vallée-Poussin and D.P. Minayeff, thought there must have been assemblies after the Buddha's death, but considered only the main characters and some events before or after the First Council historical. Other scholars, such as Bareau and Indologist Hermann Oldenberg , considered it likely that the account of

25403-412: The initial stage of sotapanna . Meant as an encouragement, the Buddha was referring to Ānanda. During the Buddha's final Nirvana , Anuruddha was able to use his meditative powers to understand which stages the Buddha underwent before attaining final Nirvana. However, Ānanda was unable to do so, indicating his lesser spiritual maturity. After the Buddha's death, Ānanda recited several verses, expressing

25594-511: The living memory of the Buddha, without which the saṅgha would be much worse off. Besides his memory skills, Ānanda also stood out in that, as the Buddha's cousin, he dared to ask the Buddha direct questions. For example, after the death of Mahāvira and the depicted subsequent conflicts among the Jain community, Ānanda asked the Buddha how such problems could be prevented after the Buddha's death. However, Findly argues that Ānanda's duty to memorize

25785-455: The main reason for his concessions with regard to the establishment of a bhikkhunī order. Many scholars interpret this account to mean that the Buddha was reluctant in allowing women to be ordained, and that Ānanda successfully persuaded the Buddha to change his mind. For example, Indologist and translator I.B. Horner wrote that "this is the only instance of his [the Buddha] being over-persuaded in argument". However, some scholars interpret

25976-459: The mango grove called Anupiya, part of Malla territory. According to a text from the Mahāsaṅghika tradition, King Suddhodana wanted the Buddha to have more followers of the khattiya caste (Sanskrit: kṣatriyaḥ , lit.   'warrior-noble, member of the ruling class'), and less from the brahmin (priest) caste. He therefore ordered that any khattiya who had a brother follow

26167-415: The metaphysical approach are now apparent. It has tried to find comprehensive answers without knowing the answers to the more restricted questions involved - such questions as those of the epistemological and logical structure of the system. Robinson (1957: p. 296) conveys his focus and states his methodology, clearly identifying the limitations in scope of this particular publication, which he testifies

26358-497: The mission following the Buddha's prediction. The latter's pupil Upagupta was described to be the teacher of King Aśoka (3rd century BCE). Together with four or five other pupils of Ānanda, Sāṇavāsī and Majjhantika formed the majority of the Second Council, with Majjhantika being Ānanda's last pupil. Post-canonical Pāli sources add that Sāṇavāsī had a leading role in the Third Buddhist Council as well. Although little

26549-408: The monks at risk for theft. To decrease its commercial value, monks therefore cut up the cloth offered, before they sew a robe from it. The Buddha asked Ānanda to think of a model for a Buddhist robe, made from small pieces of cloth. Ānanda designed a standard robe model, based on the rice fields of Magadha, which were divided in sections by banks of earth. Another tradition that is connected to Ānanda

26740-536: The mutual opposition between two views": As Shakyamuni relates in a 'thread' (Sanskrit: sūtra ) of discourse to Kaccānagotta in the Kaccānagotta Sutta , rendered into English by Myanmar Piṭaka Association Editorial Committee (1993: p. 35): "For the most part, Kaccāna, sentient beings depend on two kinds of belief - belief that 'there is' (things exist) and belief that 'there is not' (things do not exist). Y Karunadasa (1999, 2000: p. 1) states that: ...it

26931-404: The number of participants on the council, and the account of Ānanda's enlightenment just before the council. Regardless, today, the story of Ānanda's struggle on the evening before the council is still told among Buddhists as a piece of advice in the practice of meditation : neither to give up, nor to interpret the practice too rigidly. The First Council began when Ānanda was consulted to recite

27122-417: The ones that emphasized discipline; the former emphasizing the figure of Ānanda, the latter Mahākassapa. He further argued that the passage on Māra obstructing the Buddha was inserted in the fourth century BCE, and that Ānanda was blamed for Māra's doing by inserting the passage of Ānanda's forgetfulness in the third century BCE. The passage in which the Buddha was ill and reminded Ānanda to be his own refuge, on

27313-550: The only solution of f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0} is the point P 0 = ( x 0 , y 0 , z 0 ) {\displaystyle P_{0}=(x_{0},y_{0},z_{0})} and the equation is said to be the equation of a point sphere . Finally, in the case ρ > 0 {\displaystyle \rho >0} , f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0}

27504-433: The other hand, Bareau regarded as very ancient, pre-dating the passages blaming Māra and Ānanda. In conclusion, Bareau, Przyluski and Horner argued that the offenses Ānanda were charged with were a later interpolation. Findly disagrees, however, because the account in the texts of monastic discipline fits in with the Mahāparinibbāna Sutta and with Ānanda's character as generally depicted in the texts. Tradition states that

27695-470: The other hand, archaeologist Louis Finot , Indologist E. E. Obermiller and to some extent Indologist Nalinaksha Dutt thought the account of the First Council was authentic, because of the correspondences between the Pāli texts and the Sanskrit traditions . Indologist Richard Gombrich , following Bhikkhu Sujato and Bhikkhu Brahmali's arguments, states that "it makes good sense to believe ... that large parts of

27886-600: The palace. Ānanda responded to his mother's resistance by moving to Videha (Sanskrit: Vaideha ) and lived there, taking a vow of silence. This led him to gain the epithet Videhamuni (Sanskrit: Vaidehamuni ), meaning 'the silent wise one from Videha'. When Ānanda did become ordained, his father had him ordain in Kapilavatthu in the Nigrodhārāma monastery (Sanskrit: Niyagrodhārāma ) with much ceremony, Ānanda's preceptor (Pali: upajjhāya ; Sanskrit: upādhyāya ) being

28077-405: The person who was responsible for the recitation of the monastic discipline, recited before Ānanda does: again, monastic discipline above discourse. Analyzing six recensions of different textual traditions of the Mahāparinibbāna Sutta extensively, Bareau distinguished two layers in the text, an older and a newer one, the former belonging to the compilers that emphasized discourse, the latter to

28268-414: The personal needs of the Buddha, but also the needs of the still young, growing institute of the saṅgha . Moreover, because of his ability to remember the many teachings of the Buddha, he is described as foremost in "having heard much" (Pali: bahussuta , Sanskrit: bahuśruta , pinyin : Duowen Diyi ). Ānanda was known for his exceptional memory, which is essential in helping him to remember

28459-422: The poles is called the equator . Great circles through the poles are called lines of longitude or meridians . Small circles on the sphere that are parallel to the equator are circles of latitude (or parallels ). In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding. Mathematicians consider

28650-486: The present age. He met the attendant of Padumuttara Buddha and aspired to be like him in a future life. After having done many good deeds , he made his resolution known to the Padumuttara Buddha, who confirmed that his wish will come true in a future life. After having been born and reborn throughout many lifetimes, and doing many good deeds, he was born as Ānanda in the time of the current Buddha Gotama . Ānanda

28841-489: The process. Ānanda was attractive in appearance. A Pāli account related that a bhikkhunī (nun) became enamored with Ānanda, and pretended to be ill to have Ānanda visit her. When she realized the error of her ways, she confessed her mistakes to Ānanda. Other accounts relate that a low-caste woman called Prakṛti (also known in China as 摩登伽女 ; Módēngqiénǚ ) fell in love with Ānanda, and persuaded her mother Mātaṅgī to use

29032-467: The question of creation but was not used as a significant tool of logic before Buddhism. Śākyamuni , as remembered by Ānanda and codified in the Brahmajala Sutta 2.27, when expounding the sixteenth wrong view, or the fourth wrong view of the 'Eel-Wrigglers' (Pali: amarā-vikheppikā) , the non-committal equivocators who adhered to Ajñana , the sceptical philosophy, though the grammatical structure

29223-431: The river Ganges. However, according to Pāli sources, when Ānanda was about to die, he decided to spend his final moments in Vesālī instead, and traveled to the river Rohīni . The Mūlasarvāstivāda version expands and says that before reaching the river, he met with a seer called Majjhantika (following the prediction earlier) and five hundred of his followers, who converted to Buddhism. Some sources add that Ānanda passed

29414-548: The river Rohīni. Faxian also reported having visited stūpas dedicated to Ānanda at the river Rohīni, but also in Mathurā. Moreover, according to the Mūlasarvāstivāda version of the Saṃyukta Āgama , King Aśoka visited and made the most lavish offerings he ever made to a stūpa : "Who in the Norm is widely versed, He explained to his ministers that he did this because "[t]he body of

29605-417: The river, or in some versions of the account, splitting in four parts. In this way, Ānanda had pleased all the parties involved. In some other versions of the account, including the Mūlasarvāstivāda version, his death took place on a barge in the middle of the river, however, instead of in mid-air. The remains were divided in two, following the wishes of Ānanda. Majjhantika later successfully carried out

29796-455: The robe to the community as a whole rather than to be attached to his person. However, Ānanda interceded and mediated, suggesting that the Buddha had better accept the robe. Eventually the Buddha did, but not without pointing out to Ānanda that good deeds like giving should always be done for the sake of the action itself, not for the sake of the person. The texts say that the Buddha sometimes asked Ānanda to substitute for him as teacher, and

29987-451: The run-of the-mill person could identify with Ānanda, the Buddha could through Ānanda convey his teachings to the mass easily. Ānanda's character was in many ways a contradiction to that of the Buddha: being unenlightened and someone who made mistakes. At the same time, however, he was completely devoted to service to the Buddha. The Buddha is depicted in the early texts as both a father and a teacher to Ānanda, stern but compassionate. Ānanda

30178-524: The same reason. In the Pāli, there were also two parties interested, but the two parties were the Sākiyan and the Koliyan clans instead. Ānanda realized that his death on either side of the river could anger one of the parties involved. Through a supernatural accomplishment, he therefore surged into the air to levitate and meditate in mid-air, making his body go up in fire, with his relics landing on both banks of

30369-504: The scholarly lineage: "Certainly some of Nagarjuna's ancient opponents were just as confused as his modern interpreters...". This noise may also have co-arisen with Nagarjuna, following the work of Jayatilleke (1967). Robinson (1957: p. 294) holds that Stcherbatsky (1927), opened a productive period in Madhyamaka studies. Schayer (1933) made a departure into the rules of inference employed by early Buddhist dialecticians and examines

30560-424: The scriptures of each tradition: e.g. the Pāli and Mahīśāsaka textual traditions portray a Mahākassapa that is more critical of Ānanda than that the Sarvāstivāda tradition depicts him, reflecting a preference for discipline above discourse on the part of the former traditions, and a preference for discourse for the latter. Another example is the recitations during the First Council. The Pāli texts state that Upāli ,

30751-465: The sexes. The garudhamma , however, were meant to fix these problems, and prevent the dispensation from being curtailed. There are some chronological discrepancies in the traditional account of the setting up of the bhikkhunī order. According to the Pāli and Mahīśasaka textual traditions, the bhikkhunī order was set up five years after the Buddha's enlightenment, but, according to most textual traditions, Ānanda only became attendant twenty years after

30942-440: The smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area. The surface area relative to the mass of a ball is called the specific surface area and can be expressed from

31133-563: The spell of reality, a spell dispelled by the Catuṣkoṭi: We are typically not aware of ourselves as taking something (P) as real. Rather, its reality "takes us," or already has us in its spell as soon as we become aware of its identity (P). Furthermore, it's impossible to take something (P) to be real without, at least momentarily, ignoring or denying that which it is not (not-P). Thus the act of taking something as real necessarily involves some degree of unconsciousness or lack of awareness. This

31324-506: The sphere has the same center and radius as the sphere, and divides it into two equal hemispheres . Although the figure of Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. A particular line passing through its center defines an axis (as in Earth's axis of rotation ). The sphere-axis intersection defines two antipodal poles ( north pole and south pole ). The great circle equidistant to

31515-522: The sphere with radius r > 0 {\displaystyle r>0} and center ( x 0 , y 0 , z 0 ) {\displaystyle (x_{0},y_{0},z_{0})} can be parameterized using trigonometric functions . The symbols used here are the same as those used in spherical coordinates . r is constant, while θ varies from 0 to π and φ {\displaystyle \varphi } varies from 0 to 2 π . In three dimensions,

31706-522: The sphere). This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle . This formula can also be derived using integral calculus (i.e., disk integration ) to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked side by side and centered along

31897-427: The spiritual themes, Wagner also addresses the faults of the caste system by having the Buddha criticize it. Sphere A sphere (from Greek σφαῖρα , sphaîra ) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle . Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space . That given point

32088-432: The square of the distance between their centers is equal to the sum of the squares of their radii. If f ( x , y , z ) = 0 and g ( x , y , z ) = 0 are the equations of two distinct spheres then is also the equation of a sphere for arbitrary values of the parameters s and t . The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition

32279-411: The start of the bhikkhunī order. Ānanda had Mahāpajāpati ordained by her acceptance of a set of rules, set by the Buddha. These came to be known as the garudhamma , and they describe the subordinate relation of the bhikkhunī community to that of the bhikkhus or monks. Scholar of Asian religions Reiko Ohnuma argues that the debt the Buddha had toward his foster-mother Mahāpajāpati may have been

32470-412: The technical Buddhist terms 'existence', 'nature', 'being', 'entity' and 'svabhava' which are all mutually qualifying. Robinson (1957: p. 297) renders Mūlamadhyamakakārikā 21.14, thus: "He who posits an entity becomes entangled in eternalism and annihilism, since that entity has to be either permanent or impermanent." Robinson (1957: p. 300) in discussing the Buddhist logic of Nagarjuna , frames

32661-557: The text (Sanskrit: saṃgītikāra ), had first-hand experience and did not add anything to it. Thus, the discourses Ānanda remembered later became the collection of discourses of the Canon, and according to the Haimavāta , Dharmaguptaka and Sarvāstivāda textual traditions (and implicitly, post-canonical Pāli chronicles), the collection of Abhidhamma ( Abhidhamma Piṭaka ) as well. Scholar of religion Ronald Davidson notes, however, that this

32852-418: The texts he is depicted as compassionate in his relations with lay people, a compassion he learnt from the Buddha. The Buddha relays that both monastics and lay people were pleased to see Ānanda, and were pleased to hear him recite and teach the Buddha's teaching. Moreover, Ānanda was known for his organizational skills, assisting the Buddha with secretary-like duties. In many ways, Ānanda did not only serve

33043-449: The two, as indicated in the early texts. Regardless, it is clear from the texts that a relationship of transmission of teachings is meant, as opposed to an upajjhāya –student relationship in a lineage of ordination: no source indicates Mahākassapa was Ānanda's upajjhāya . In Mahāyāna iconography, Ānanda is often depicted flanking the Buddha at the right side, together with Mahākassapa at the left. In Theravāda iconography, however, Ānanda

33234-418: The Ānanda Bodhi Tree is sometimes identified with a tree at the ruins of Jetavana , Sāvatthi, based on the records of Faxian. Between 1856 and 1858 Richard Wagner wrote a draft for an opera libretto based on the legend about Ānanda and the low-caste girl Prakṛti. He left only a fragmentary prose sketch of a work to be called Die Sieger , but the topic inspired his later opera Parsifal . Furthermore,

33425-477: Was 35 years old), and was therefore much younger than the Buddha. The latter tradition is corroborated by several instances in the Early Buddhist Texts, in which Ānanda appears younger than the Buddha, such as the passage in which the Buddha explained to Ānanda how old age was affecting him in body and mind. It is also corroborated by a verse in the Pāli text called Theragāthā , in which Ānanda stated he

33616-497: Was a "learner" for 25 years, after which he attended to the Buddha for another 25 years. Following the Pāli , Mahīśasaka and Dharmaguptaka textual traditions, Ānanda became a monk in the second year of the Buddha's ministry, during the Buddha's visit to Kapilavatthu (Sanskrit: Kapilavastu ). He was ordained by the Buddha himself, together with many other princes of the Buddha's clan (Pali: Sākiya , Sanskrit: Śākya ), in

33807-399: Was a rule issued that only enlightened disciples ( arahants ) were allowed to attend the council, to prevent mental afflictions from clouding the disciples' memories. Ānanda had, however, not attained enlightenment yet, in contrast with the rest of the council, consisting of 499 arahants . Mahākassapa therefore did not allow Ānanda to attend yet. Although he knew that Ānanda's presence in

33998-483: Was born in the same time period as the Buddha (formerly Prince Siddhattha), which scholars place at 5th–4th centuries BCE. Tradition says that Ānanda was the first cousin of the Buddha, his father being the brother of Suddhodana (Sanskrit: Śuddhodana ), the Buddha's father. In the Pāli and Mūlasarvāstivāda textual traditions, his father was Amitodana (Sanskrit: Amṛtodana ), but the Mahāvastu states that his father

34189-533: Was born, his relatives were joyous about this. Texts from the Mūlasarvāstivāda tradition, however, state that since Ānanda was born on the day of the Buddha's enlightenment, there was great rejoicing in the city—hence the name. According to the texts, in a previous life, Ānanda made an aspiration to become a Buddha 's attendant. He made this aspiration in the time of a previous Buddha called Padumuttara , many eons (Pali: kappa , Sanskrit: kalpa ) before

34380-408: Was carefully used, reused and recycled by the monastic community, prompting the king to offer another five hundred robes. Ānanda also had a role in the Buddha's visit to Vesālī . In this story, the Buddha taught the well-known text Ratana Sutta to Ānanda, which Ānanda then recited in Vesālī, ridding the city from illness, drought and evil spirits in the process. Another well-known passage in which

34571-412: Was criticized by the monk Mahākassapa , several bhikkhunīs tried to defend him. According to Indologist Oskar von Hinüber , Ānanda's pro- bhikkhunī attitude may well be the reason why there was frequent discussion between Ānanda and Mahākassapa, eventually leading Mahākasapa to charge Ānanda with several offenses during the First Buddhist Council. Von Hinüber further argues that the establishment of

34762-534: Was employed particularly by Nagarjuna who developed it and engaged it as a 'learning, investigative, meditative' portal to realize the 'openness' (Sanskrit: Śūnyatā ), of Shakyamuni's Second Turning of the Dharmacakra , as categorized by the Sandhinirmocana Sutra . Robinson (1957: p. 294), building on the foundations of Liebenthal (1948) to whom he gives credit, states: What Nagarjuna wishes to prove

34953-515: Was known as the disciple who attained awakening "in none of the four traditional poses " (walking, standing, sitting, or lying down). The next morning, to prove his enlightenment, Ānanda performed a supernatural accomplishment by diving into the earth and appearing on his seat at the council (or, according to some sources, by flying through the air). Scholars such as Buddhologist André Bareau and scholar of religion Ellison Banks Findly have been skeptical about many details in this account, including

35144-439: Was less experienced in meditative concentration than other leading disciples. Thus, judgment of Ānanda's character depends on whether one judges his accomplishments as a monk or his accomplishments as an attendant, and person memorizing the discourses. From a literary and didactic point of view, Ānanda often functioned as a kind of foil in the texts, being an unenlightened disciple attending to an enlightened Buddha. Because

35335-470: Was not properly dressed, and during which his body was sullied by their tears; and for having failed to ask the Buddha to continue to live on. Ānanda did not acknowledge these as offenses, but he conceded to do a formal confession anyway, "... in faith of the opinion of the venerable elder monks" —Ānanda wanted to prevent disruption in the saṅgha . With regard to having women ordained, Ānanda answered that he had done this with great effort, because Mahāpajāpati

35526-672: Was not yet enlightened, as opposed to the Buddha. In the Sanskrit textual traditions , Ānanda is considered the patriarch of the Dhamma who stood in a spiritual lineage, receiving the teaching from Mahākassapa and passing them on to his own pupils. Ānanda has been honored by bhikkhunīs since early medieval times for his merits in establishing the nun's order. In recent times, the composer Richard Wagner and Indian poet Rabindranath Tagore were inspired by stories about Ānanda in their work. The word ānanda (आनंद) means 'bliss, joy' in Pāli and in Sanskrit. Pāli commentaries explain that when Ānanda

35717-544: Was often praised by the Buddha for his teachings. Ānanda was often given important teaching roles, such as regularly teaching Queen Mallikā , Queen Sāmāvatī , (Sanskrit: Śyāmāvatī ) and other people from the ruling class. Once Ānanda taught a number of King Udena (Sanskrit: Udayana )'s concubines. They were so impressed by Ānanda's teaching, that they gave him five hundred robes, which Ānanda accepted. Having heard about this, King Udena criticized Ānanda for being greedy; Ānanda responded by explaining how every single robe

35908-399: Was the Buddha's foster-mother who had long provided for him. With regard to not requesting the Buddha to continue to live, many textual traditions have Ānanda respond by saying he was distracted by Māra, though one early Chinese text has Ānanda reply he did not request the Buddha to prolong his life, for fear that this would interfere with the next Buddha Maitreya 's ministry. According to

36099-405: Was the primary attendant of the Buddha and one of his ten principal disciples . Among the Buddha's many disciples, Ānanda stood out for having the best memory. Most of the texts of the early Buddhist Sutta-Piṭaka ( Pali : सुत्त पिटक ; Sanskrit : सूत्र-पिटक , Sūtra-Piṭaka ) are attributed to his recollection of the Buddha's teachings during the First Buddhist Council . For that reason, he

36290-412: Was very fond of and attached to the Buddha, willing to give his life for him. He mourned the deaths of both the Buddha and Sāriputta, with whom he enjoyed a close friendship: in both cases Ānanda was very shocked. Ānanda's faith in the Buddha, however, constituted more of a faith in a person, especially the Buddha's person, as opposed to faith in the Buddha's teaching. This is a pattern which comes back in

36481-447: Was Śuklodana—both are brothers of Suddhodana. The Mahāvastu also mentions that Ānanda's mother's name was Mṛgī (Sanskrit; lit. 'little deer'; Pāli is unknown). The Pāli tradition has it that Ānanda was born on the same day as Prince Siddhatta (Sanskrit: Siddhārtha ), but texts from the Mūlasarvāstivāda and subsequent Mahāyāna traditions state Ānanda was born at the same time the Buddha attained enlightenment (when Prince Siddhattha

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