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Book of Daniel
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The Book of Daniel is a biblical apocalypse, combining a prophecy of history with an eschatology (the study of last things) which is both cosmic in scope and political in its focus.[1] In more mundane language, it is "an account of the activities and visions of Daniel, a noble Jew exiled at Babylon,"[2] its message being that just as the God of Israel saved Daniel and his friends from their enemies, so he would save all of Israel in their present oppression.[3]
In the Hebrew Bible it is found in the Ketuvim (writings), while in Christian Bibles it is grouped with the Major Prophets.[4] The book divides into two parts, a set of six court tales in chapters 1–6 followed by four apocalyptic visions in chapters 7–12.[5] The deuterocanon contains three additional stories, the Song of the Three Holy Children, Susanna, and Bel and the Dragon.[6]
Though the book is traditionally ascribed to Daniel himself, modern scholarly consensus considers it pseudonymous, the stories of the first half legendary in origin, and the visions of the second the product of anonymous authors in the Maccabean period (2nd century BC).[5] Its influence has resonated through later ages, from the Dead Sea Scrolls community and the authors of the gospels and Revelation, to various movements from the 2nd century to the Protestant Reformation and modern millennialist movements—on which it continues to have a profound influence.[7]
Contents
[hide]
1 Structure
1.1 Divisions
1.2 Chiastic structure in the Aramaic section
2 Content
2.1 Introduction in Babylon (chapter 1)
2.2 Nebuchadnezzar's dream of four kingdoms (chapter 2)
2.3 The fiery furnace (chapter 3)
2.4 Nebuchadnezzar's madness (chapter 4)
2.5 Belshazzar's feast (chapter 5)
2.6 Daniel in the lions' den (chapter 6)
2.7 Vision of the beasts from the sea (chapter 7)
2.8 Vision of the ram and goat (chapter 8)
2.9 Vision of the Seventy Weeks (chapter 9)
2.10 Vision of the kings of north and south (chapters 10–12)
2.11 Additions to Daniel (Greek text tradition)
3 Historical background
4 Composition
4.1 Development
4.2 Authorship
4.3 Dating
5 Manuscripts
6 Genre, meaning, symbolism and chronology
6.1 Genre
6.2 Meaning, symbolism and chronology
7 Influence
7.1 Religion
7.2 Western culture
8 See also
9 Notes
10 References
10.1 Citations
10.2 Bibliography
11 External links
Structure[edit]
Nebuchadnezzar's dream: the composite statue (France, 15th century).
Divisions[edit]
The Book of Daniel is divided between the court tales of chapters 1–6 and the apocalyptic visions of 7–12, and between the Hebrew of chapters 1 and 8–12 and the Aramaic of chapters 2–7.[8][9] The division is reinforced by the chiastic arrangement of the Aramaic chapters (see below), and by a chronological progression in chapters 1–6 from Babylonian to Median times, and from Babylonian to Persian in chapters 7–12.[10] Various suggestions have been made by scholars to explain the fact that the genre division does not coincide with the other two, but it appears that the language division and concentric structure of chapters 2–6 are artificial literary devices designed to bind the two halves of the book together.[10] The following outline is provided by Collins in his commentary on Daniel:[11]
PART I: Tales (chapters 1:1–6:29)
1: Introduction (1:1–21 – set in the Babylonian era, written in Hebrew)
2: Nebuchadnezzar's dream of four kingdoms (2:1–49 – Babylonian era; Aramaic)
3: The fiery furnace (3:1–30 – Babylonian era; Aramaic)
4: Nebuchadnezzar's madness (3:31–4:34 – Babylonian era; Aramaic)
5: Belshazzar's feast (5:1–6:1 – Babylonian era; Aramaic)
6: Daniel in the lions' den (6:2–29 – Median era with mention of Persia; Aramaic)
PART II: Visions (chapters 7:1–12:13)
7: The beasts from the sea (7:1–28 – Babylonian era: Aramaic)
8: The ram and the he-goat (8:1–27 – Babylonian era; Hebrew)
9: Interpretation of Jeremiah's prophecy of the seventy weeks (9:1–27 – Median era; Hebrew)
10: The angel's revelation: kings of the north and south (10:1–12:13 – Persian era, mention of Greek era; Hebrew)
Chiastic structure in the Aramaic section[edit]
There is a clear chiasm (a concentric literary structure in which the main point of a passage is placed in the centre and framed by parallel elements on either side in "ABBA" fashion) in the chapter arrangement of the Aramaic section. The following is taken from Paul Redditt's "Introduction to the Prophets":[12]
A1 (2:4b-49) – A dream of four kingdoms replaced by a fifth
B1 (3:1–30) – Daniel's three friends in the fiery furnace
C1 (4:1–37) – Daniel interprets a dream for Nebuchadnezzar
C2 (5:1–31) – Daniel interprets the handwriting on the wall for Belshazzar
B2 (6:1–28) – Daniel in the lions' den
A2 (7:1–28) – A vision of four world kingdoms replaced by a fifth
Content[edit]
Chapters of the Book of Daniel
Chapter 1: Induction into Babylon
Chapter 2: Nebuchadnezzar's Dream
Chapter 3: The Fiery Furnace
Chapter 4: Nebuchadnezzar's Madness
Chapter 5: Belshazzar's Feast
Chapter 6: Daniel in the Lions's Den
Chapter 7: The Four Beasts
Chapter 8: The Ram, He-Goat and Horn
Chapter 9: The Seventy Weeks
Chapters 10-12: Daniel's final vision
Additions to Daniel
Song of the Three Holy Children
Susanna and the Elders
Bel and the Dragon
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Introduction in Babylon (chapter 1)[edit]
In the third year of King Jehoiakim, God allows Jerusalem to fall into the power of Nebuchadnezzar II, king of Babylon.[Notes 1] Young Israelites of noble and royal family, "without physical defect, and handsome," versed in wisdom and competent to serve in the palace of the king, are taken to Babylon to be taught the literature and language of that nation. Among them are Daniel and his three companions, who refuse to touch the royal food and wine. Their overseer fears for his life in case the health of his charges deteriorates, but Daniel suggests a trial and the four emerge healthier than their counterparts from ten days of nothing but vegetables and water. They are allowed to continue to refrain from eating the king's food, and to Daniel God gives insight into visions and dreams. When their training is done Nebuchadnezzar finds them 'ten times better' than all the wise men in his service and therefore keeps them at his court, where Daniel continues until the first year of King Cyrus.[13][Notes 2]
Nebuchadnezzar's dream of four kingdoms (chapter 2)[edit]
Main article: Daniel 2
In the second year of his reign Nebuchadnezzar has a dream. When he wakes up, he realizes that the dream has some important message, so he consults his wise men. Wary of their potential to fabricate an explanation the king refuses to tell the wise men what he saw in his dream. Rather, he demands that his wise men tell him what the content of the dream was, and then interpret it. When the wise men protest that this is beyond the power of any man, he sentences all, including Daniel and his friends, to death. Daniel receives an explanatory vision from God: Nebuchadnezzar had seen an enormous statue with a head of gold, breast and arms of silver, belly and thighs of bronze, legs of iron, and feet of mixed iron and clay, then saw the statue destroyed by a rock that turned into a mountain filling the whole earth. Daniel explains the dream to the king: the statue symbolized four successive kingdoms, starting with Nebuchadnezzar, all of which would be crushed by God's kingdom, which would endure forever. Nebuchadnezzar acknowledges the supremacy of Daniel's god, raises Daniel over all his wise men, and places Daniel and his companions over the province of Babylon.[14]
The fiery furnace (chapter 3)[edit]
Main article: Shadrach, Meshach, and Abednego
Daniel's companions Shadrach, Meshach, and Abednego refuse to bow to King Nebuchadnezzar's golden statue and are thrown into a fiery furnace. Nebuchadnezzar is astonished to see a fourth figure in the furnace with the three, one "with the appearance like a son of the gods." So the king called the three to come out of the fire, and blessed the God of Israel, and decreed that any who blasphemed against him should be torn limb from limb.[15]
Nebuchadnezzar's madness (chapter 4)[edit]
Main article: Daniel 4
Nebuchadnezzar by William Blake (between c. 1795 and 1805)
Nebuchadnezzar recounts a dream of a huge tree that is suddenly cut down at the command of a heavenly messenger. Daniel is summoned and interprets the dream. The tree is Nebuchadnezzar himself, who for seven years will lose his mind and live like a wild beast. All of this comes to pass until, at the end of the specified time, Nebuchadnezzar acknowledges that "heaven rules" and his kingdom and sanity are restored.[16]
Belshazzar's feast (chapter 5)[edit]
Main article: Belshazzar's feast
See also: Fall of Babylon
Belshazzar and his nobles blasphemously drink from sacred Jewish temple vessels, offering praise to inanimate gods, until a hand mysteriously appears and writes upon the wall. The horrified king summons Daniel, who upbraids him for his lack of humility before God and interprets the message: Belshazzar's kingdom will be given to the Medes and Persians. Belshazzar rewards Daniel and raises him to be third in the kingdom, and that very night Belshazzar is slain and Darius the Mede takes the kingdom.[17][Notes 3]
Daniel in the lions' den (chapter 6)[edit]
Daniel's Answer to the King by Briton Rivière (1892)
Main article: Daniel in the lions' den
Darius elevates Daniel to high office, exciting the jealousy of other officials. Knowing of Daniel's devotion to his God, his enemies trick the king into issuing an edict forbidding worship of any other god or man for a 30-day period. Daniel continues to pray three times a day to God towards Jerusalem; he is accused and King Darius, forced by his own decree, throws Daniel into the lions' den. But God shuts up the mouths of the lions, and the next morning Darius rejoices to find him unharmed. The king casts Daniel's accusers into the lions' pit together with their wives and children to be instantly devoured, while he himself acknowledges Daniel's God as he whose kingdom shall never be destroyed.[18]
Vision of the beasts from the sea (chapter 7)[edit]
Main article: Daniel 7
See also: Four kingdoms of Daniel
In the first year of Belshazzar Daniel has a dream of four monstrous beasts arising from the sea.[Notes 4] The fourth, a beast with ten horns, devours the whole earth, treading it down and crushing it, and a further small horn appears and uproots three of the earlier horns. The Ancient of Days judges and destroys the beast, and "one like a son of man" is given everlasting kingship over the entire world. A divine being explains that the four beasts represent four kings, but that "the holy ones of the Most High" would receive the everlasting kingdom. The fourth beast would be a fourth kingdom with ten kings, and another king who would pull down three kings and make war on the "holy ones" for "a time, two times and a half," after which the heavenly judgement will be made against him and the "holy ones" will receive the everlasting kingdom.[19]
Vision of the ram and goat (chapter 8)[edit]
Main article: Daniel 8
In the third year of Belshazzar Daniel has vision of a ram and goat. The ram has two mighty horns, one longer than the other, and it charges west, north and south, overpowering all other beasts. A goat with a single horn appears from the west and destroys the ram. The goat becomes very powerful until the horn breaks off and is replaced by four lesser horns. A small horn that grows very large, it stops the daily temple sacrifices and desecrates the sanctuary for two thousand three hundred "evening and mornings" (which could be either 1150 or 2300 days) until the temple is cleansed. The angel Gabriel informs him that the ram represents the Medes and Persians, the goat is Greece, and the "little horn" is a wicked king.[20]
Vision of the Seventy Weeks (chapter 9)[edit]
Main article: Prophecy of Seventy Weeks
In the first year of Darius the Mede, Daniel meditates on the word of Jeremiah that the desolation of Jerusalem would last seventy years; he confesses the sin of Israel and pleads for God to restore Israel and the "desolated sanctuary" of the Temple. The angel Gabriel explains that the seventy years stand for seventy "weeks" of years (490 years), during which the Temple will first be restored, then later defiled by a "prince who is to come," "until the decreed end is poured out."[21]
Vision of the kings of north and south (chapters 10–12)[edit]
Main article: Daniel's final vision
Daniel 10: In the third year of Cyrus[Notes 5] Daniel sees in his vision an angel (called "a man", but clearly a supernatural being) who explains that he is in the midst of a war with the "prince of Persia", assisted only by Michael, "your prince." The "prince of Greece" will shortly come, but first he will reveal what will happen to Daniel's people.
Daniel 11: A future king of Persia will make war on the king of Greece, a "mighty king" will arise and wield power until his empire is broken up and given to others, and finally the king of the south (identified in verse 8 as Egypt) will go to war with the "king of the north." After many battles (described in great detail) a "contemptible person" will become king of the north; this king will invade the south two times, the first time with success, but on his second he will be stopped by "ships of Kittim." He will turn back to his own country, and on the way his soldiers will desecrate the Temple, abolish the daily sacrifice, and set up the abomination of desolation. He will defeat and subjugate Libya and Egypt, but "reports from the east and north will alarm him," and he will meet his end "between the sea and the holy mountain."
Daniel 12: At this time Michael will come. It will be a time of great distress, but all those whose names are written will be delivered. "Multitudes who sleep in the dust of the earth will awake, some to everlasting life, others to shame and everlasting contempt; those who are wise will shine like the brightness of the heavens, and those who lead many to righteousness, like the stars for ever and ever." In the final verses the remaining time to the end is revealed: "a time, times and half a time" (three years and a half). Daniel fails to understand and asks again what will happen, and is told: "From the time that the daily sacrifice is abolished and the abomination that causes desolation is set up, there will be 1,290 days. Blessed is the one who waits for and reaches the end of the 1,335 days."
Additions to Daniel (Greek text tradition)[edit]
Susanna and the Elders by Guido Reni (1820–1825)
Main article: Additions to Daniel
See also: Deuterocanonical books
The Greek text of Daniel is considerably longer than the Hebrew, due to three additional stories: they were accepted by all branches of Christianity until the Protestant movement rejected them in the 16th century on the basis that they were absent from Hebrew Bibles, but remain in Catholic and Orthodox Bibles.[22]
The Prayer of Azariah and Song of the Three Holy Children, placed after Daniel 3:23;
The story of Susanna and the Elders, placed before chapter 1 in some Greek versions and after chapter 12 in others;
The story of Bel and the Dragon, placed at the end of the book.
Historical background[edit]
Daniel refusing to eat at the King's table, early 1900s Bible illustration
The visions of chapters 7–12 reflect the crisis which took place in Judea in 167–164 BC when Antiochus IV Epiphanes, the Greek king of the Seleucid Empire, threatened to destroy traditional Jewish worship in Jerusalem.[23] When Antiochus came to the throne the Jews were largely pro-Seleucid. The High Priestly family was split by rivalry, and one member, Jason, offered the king a large sum to be made High Priest. Jason also asked—or more accurately, paid—to be allowed to make Jerusalem a polis, or Greek city. This meant, among other things, that city government would be in the hands of the citizens, which meant in turn that citizenship would be a valuable commodity, to be purchased from Jason. None of this threatened the Jewish religion, and the reforms were widely welcomed, especially among the Jerusalem aristocracy and the leading priests. Three years later Jason was deposed when another priest, Menelaus, offered Antiochus an even larger sum for the post of High Priest.[24]
Antiochus invaded Egypt twice, in 169 BC with success, but on the second incursion, in late 168 BC, he was forced to withdraw by the Romans.[25] Jason, hearing a rumour that Antiochus was dead, attacked Menelaus to take back the High Priesthood.[25] Antiochus drove Jason out of Jerusalem, plundered the Temple, and introduced measures to pacify his Egyptian border by imposing complete Hellenisation: the Jewish Book of the Law was prohibited and on 15 December 167 BC an "abomination of desolation", probably a Greek altar, was introduced into the Temple.[26] With the Jewish religion now clearly under threat a resistance movement sprang up, led by the Maccabee brothers, and over the next three years it won sufficient victories over Antiochus to take back and purify the Temple.[25]
The crisis which the author of Daniel addresses is the defilement of the altar in Jerusalem in 167 BC (first introduced in chapter 8:11): the daily offering which used to take place twice a day, at morning and evening, stopped, and the phrase "evenings and mornings" recurs through the following chapters as a reminder of the missed sacrifices.[27] But whereas the events leading up to the sacking of the Temple in 167 BC and the immediate aftermath are remarkably accurate (chapter 11:21–29), the predicted war between the Syrians and the Egyptians (11:40–43) never took place, and the prophecy that Antiochus would die in Palestine (11:44–45) was inaccurate (he died in Persia).[28] The conclusion is that the account must have been completed near the end of the reign of Antiochus but before his death in December 164 BC, or at least before news of it reached Jerusalem.[29]
Composition[edit]
Nebuchadnezzar's dream: the felled tree (France, 15th century).
Development[edit]
It is generally accepted that Daniel originated as a collection of Aramaic court tales later expanded by the Hebrew revelations.[30] The court tales may have originally circulated independently, but the edited collection was probably composed in the third or early second century BC.[31] When the full collection was assembled, it is likely that the brief Aramaic introduction of chapter 1 was composed to provide historical context, introduce the characters of the tales, and explain how Daniel and his friends came to Babylon.[32] In the third stage the visions of chapters 7–12 were added and chapter 1 was translated into Hebrew.[32]
Authorship[edit]
Daniel is one of a large number of Jewish apocalypses, all of them pseudonymous.[33] Although the entire book is traditionally ascribed to Daniel the seer, chapters 1–6 are in the voice of an anonymous narrator, except for chapter 4 which is in the form of a letter from king Nebuchadnezzar; only the second half (chapters 7–12) is presented by Daniel himself, introduced by the anonymous narrator in chapters 7 and 10.[34] The real author/editor of Daniel was probably an educated Jew, knowledgeable in Greek learning, and of high standing in his own community. The book is a product of "Wisdom" circles, but the type of wisdom is mantic (the discovery of heavenly secrets from earthly signs) rather than the wisdom of learning—the main source of wisdom in Daniel is God's revelation.[35][36]
It is possible that the name of Daniel was chosen for the hero of the book because of his reputation as a wise seer in Hebrew tradition.[37] Ezekiel, who lived during the Babylonian exile, mentioned him in association with Noah and Job (Ezekiel 14:14) as a figure of legendary wisdom (28:3), and a hero named Daniel (more accurately Dan'el, but the spelling is close enough for the two to be regarded as identical) features in a late 2nd millennium myth from Ugarit.[38] "The legendary Daniel, known from long ago but still remembered as an exemplary character ... serves as the principal human "hero" in the biblical book that now bears his name"; Daniel is the wise and righteous intermediary who is able to interpret dreams and thus convey the will of God to humans, the recipient of visions from on high that are interpreted to him by heavenly intermediaries.[39]
Dating[edit]
The prophecies of Daniel are accurate down to the career of Antiochus IV Epiphanes, king of Syria and oppressor of the Jews, but not in its prediction of his death: the author seems to know about Antiochus' two campaigns in Egypt (169 and 167 BC), the desecration of the Temple (the "abomination of desolation"), and the fortification of the Akra (a fortress built inside Jerusalem), but he seems to know nothing about the reconstruction of the Temple or about the actual circumstances of Antiochus' death in late 164 BC. Chapters 10–12 must therefore have been written between 167 and 164 BC. There is no evidence of a significant time lapse between those chapters and chapters 8 and 9, and chapter 7 may have been written just a few months earlier again.[40]
Further evidence of the book's date is in the fact that Daniel is excluded from the Hebrew Bible's canon of the prophets, which was closed around 200 BC, and the Wisdom of Sirach, a work dating from around 180 BC, draws on almost every book of the Old Testament except Daniel, leading scholars to suppose that its author was unaware of it. Daniel is, however, quoted in a section of the Sibylline Oracles commonly dated to the middle of the 2nd century BC, and was popular at Qumran at much the same time, suggesting that it was known from the middle of that century.[41]
Manuscripts[edit]
The Book of Daniel is preserved in the 12-chapter Masoretic Text and in two longer Greek versions, the original Septuagint version, c. 100 BC, and the later Theodotion version from c. 2nd century AD. Both Greek texts contain three additions to Daniel: The Prayer of Azariah and Song of the Three Holy Children; the story of Susannah and the Elders; and the story of Bel and the Dragon. Theodotion is much closer to the Masoretic Text and became so popular that it replaced the original Septuagint version in all but two manuscripts of the Septuagint itself.[42][43][8] The Greek additions were apparently never part of the Hebrew text.[44]
Eight copies of the Book of Daniel, all incomplete, have been found at Qumran, two in Cave 1, five in Cave 4, and one in Cave 6. Between them, they preserve text from eleven of Daniel's twelve chapters, and the twelfth is quoted in the Florilegium (a compilation scroll) 4Q174, showing that the book at Qumran did not lack this conclusion. All eight manuscripts were copied between 125 BC (4QDanc) and about 50 AD (4QDanb), showing that Daniel was being read at Qumran only about 40 years after its composition. All appear to preserve the 12-chapter Masoretic version rather than the longer Greek text. None reveal any major disagreements against the Masoretic, and the four scrolls that preserve the relevant sections (1QDana, 4QDana, 4QDanb, and 4QDand) all follow the bilingual nature of Daniel where the book opens in Hebrew, switches to Aramaic at 2:4b, then reverts to Hebrew at 8:1.[45]
Genre, meaning, symbolism and chronology[edit]
Daniel in the lions' den saved by Habakkuk (France, 15th century).
(This section deals with modern scholarly reconstructions of the meaning of Daniel to its original authors and audience)
Genre[edit]
The Book of Daniel is an apocalypse, a literary genre in which a heavenly reality is revealed to a human recipient; such works are characterized by visions, symbolism, an other-worldly mediator, an emphasis on cosmic events, angels and demons, and pseudonymity (false authorship).[46] The production of apocalypses occurred commonly from 300 BC to 100 AD, not only among Jews and Christians, but also among Greeks, Romans, Persians and Egyptians.[47] Daniel, the book's hero, is a representative apocalyptic seer, the recipient of divine revelation: he has learned the wisdom of the Babylonian magicians and surpassed them, because his God is the true source of knowledge; he is one of the maskilim (משכלים), the wise ones, who have the task of teaching righteousness and whose number may be considered to include the authors of the book itself.[47] The book is also an eschatology, as the divine revelation concerns the end of the present age, a predicted moment in which God will intervene in history to usher in the final kingdom.[48] It gives no real details of the end-time, but it seems that God's kingdom will be on this earth, that it will be governed by justice and righteousness, and that the tables will be turned on the Seleucids and those Jews who have cooperated with them.[49]
Meaning, symbolism and chronology[edit]
The message of the Book of Daniel is that, just as the God of Israel saved Daniel and his friends from their enemies, so he would save all Israel in their present oppression.[3] The book is filled with monsters, angels, and numerology, drawn from a wide range of sources, both biblical and non-biblical, that would have had meaning in the context of 2nd century Jewish culture, and while Christian interpreters have always viewed these as predicting events in the New Testament—"the Son of God", "the Son of Man", Christ and the Antichrist—the book's intended audience is the Jews of the 2nd century BC.[50] The following explains a few of these predictions, as understood by modern biblical scholars.
The four kingdoms and the little horn (Daniel 2 and 7): The concept of four successive world empires stems from Greek theories of mythological history;[51] most modern interpreters agree that the four represent Babylon, the Medes, Persia and the Greeks, ending with Hellenistic Seleucid Syria and with Hellenistic Ptolemaic Egypt.[52] The symbolism of four metals in the statue in chapter 2 comes from Persian writings,[51] while the four "beasts from the sea" in chapter 7 reflect Hosea 13:7–8, in which God threatens that he will be to Israel like a lion, a leopard, a bear or a wild beast.[53] The consensus among scholars is that the four beasts of chapter 7, like the metals of chapter 2, symbolise Babylon, Media, Persia and the Seleucids, with Antiochus IV (reigned 175–164 BC) as the "small horn" that uproots three others (Antiochus usurped the rights of several other claimants to become king of the Seleucid Empire).[54]
The Ancient of Days and the one like a son of man (Daniel 7): The portrayal of God in Daniel 7:13 resembles the portrayal of the Canaanite god El as an ancient divine king presiding over the divine court.[55] The "Ancient of Days" gives dominion over the earth to "one like a son of man", and then in Daniel 7:27 to "the people of the holy ones of the Most High", whom scholars consider the son of man to represent. These people can be understood as the maskilim (sages), or as the Jewish people broadly.[56][Notes 6]
The ram and he-goat (Daniel 8) as conventional astrological symbols represent Persia and Syria, as the text explains. The "mighty horn" stands for Alexander the Great (reigned 336–323 BC) and the "four lesser horns" represent the four principal generals (Diadochi) who fought over the Greek empire following Alexander's death. The "little horn" again represents Antiochus IV. The key to the symbols lies in the description of the little horn's actions: he ends the continual burnt offering and overthrows the Sanctuary, a clear reference to Antiochus' desecration of the Temple.[57]
The anointed ones and the seventy years (Chapter 9): Daniel reinterprets Jeremiah's "seventy years" prophecy regarding the period Israel would spend in bondage to Babylon. From the point of view of the Maccabean era, Jeremiah's promise was obviously not true—the gentiles still oppressed the Jews, and the "desolation of Jerusalem" had not ended. Daniel therefore reinterprets the seventy years as seventy "weeks" of years, making up 490 years. The 70 weeks/490 years are subdivided, with seven "weeks" from the "going forth of the word to rebuild and restore Jerusalem" to the coming of an "anointed one", while the final "week" is marked by the violent death of another "anointed one", probably the High Priest Onias III (ousted to make way for Jason and murdered in 171 BC), and the profanation of the Temple. The point of this for Daniel is that the period of gentile power is predetermined, and is coming to an end.[58][59]
Kings of north and south: Chapters 10 to 12 concern the war between these kings, the events leading up to it, and its heavenly meaning. In chapter 10 the angel (Gabriel?) explains that there is currently a war in heaven between Michael, the angelic protector of Israel, and the "princes" (angels) of Persia and Greece; then, in chapter 11, he outlines the human wars which accompany this—the mythological concept sees standing behind every nation a god/angel who does battle on behalf of his people, so that earthly events reflect what happens in heaven. The wars of the Ptolemies ("kings of the south") against the Seleucids ("kings of the north") are reviewed down to the career of Antiochus the Great (Antiochus III (reigned 222–187 BC), father of Antiochus IV), but the main focus is Antiochus IV, to whom more than half the chapter is devoted. The accuracy of these predictions lends credibility to the real prophecy with which the passage ends, the death of Antiochus—which, in the event, was not accurate.[60]
Predicting the end-time (Daniel 8:14 and 12:7–12): Biblical eschatology does not generally give precise information as to when the end will come,[61] and Daniel's attempts to specify the number of days remaining is a rare exception.[62] Daniel asks the angel how long the "little horn" will be triumphant, and the angel replies that the Temple will be reconsecrated after 2300 "evenings and mornings" have passed (Daniel 8:14). The angel is counting the two daily sacrifices, so the period is 1150 days from the desecration in December 167. In chapter 12 the angel gives three more dates: the desolation will last "for a time, times and half a time", or a year, two years, and a half a year (Daniel 12:8); then that the "desolation" will last for 1290 days (12:11); and finally, 1335 days (12:12). Verse 12:11 was presumably added after the lapse of the 1150 days of chapter 8, and 12:12 after the lapse of the number in 12:11.[63]
Influence[edit]
Engraving of Daniel's vision of the four beasts in chapter 7 by Matthäus Merian, 1630.
Religion[edit]
The concepts of immortality and resurrection, with rewards for the righteous and punishment for the wicked, have roots much deeper than Daniel, but the first clear statement is found in the final chapter of that book: "Many of those who sleep in the dust of the earth shall awake, some to everlasting life, and some to everlasting shame and contempt."[64] Without this belief, Christianity, in which the resurrection of Jesus plays a central role, would have disappeared, like the movements following other charismatic Jewish figures of the 1st century.[65]
Daniel was quoted and referenced by both Jews and Christians in the 1st century AD as predicting the imminent end-time.[66] Moments of national and cultural crisis continually reawakened the apocalyptic spirit, through the Montanists of the 2nd/3rd centuries, persecuted for their millennialism, to the more extreme elements of the 16th century Reformation such as the Zwickau prophets and the Münster Rebellion.[7] During the English Civil War, the Fifth Monarchy Men took their name and political program from Daniel 7, demanding that Oliver Cromwell allow them to form a "government of saints" in preparation for the coming of the Messiah; when Cromwell refused, they identified him instead as the Beast usurping the rightful place of King Jesus.[67] Daniel remains one of the most influential apocalypses in modern America, foretelling the history of Jesus and the Second Coming.[68]
The influence of Daniel has not been confined to Judaism and Christianity: In the Middle Ages Muslims created horoscopes whose authority was attributed to Daniel. More recently the Bahá'í movement, which originated in Persian Shi'ite Islam, justified its existence on the 1260-day prophecy of Daniel, holding that it foretold the coming of the Twelfth Imam and an age of peace and justice in the year 1844, which is the year 1260 of the Muslim era.[69]
Western culture[edit]
Daniel belongs not only to the religious tradition but also to the wider Western intellectual and artistic heritage. It was easily the most popular of the prophetic books for the Anglo-Saxons, who nevertheless treated it not as prophecy but as a historical book, "a repository of dramatic stories about confrontations between God and a series of emperor-figures who represent the highest reach of man".[70] In the early modern period the physicist Isaac Newton paid special attention to it, and Francis Bacon borrowed a motto from it for his work Novum Organum. Philosophers, such as Baruch Spinoza drew on it. In the 20th century its apocalyptic second half attracted the attention of Carl Jung. The book has also inspired musicians, from medieval liturgical drama to the 20th century compositions of Darius Milhaud. Artists including Michelangelo, Rembrandt and Eugène Delacroix have all drawn on its imagery.[69]
See also[edit]
Biblical numerology
Christian eschatology
Daniel (Old English poem)
Greek Apocalypse of Daniel
Notes[edit]
Jump up ^ Jehoiakim: King of Judah 608–598 BC; his third year would be either 606 or 605, depending how years are counted.
Jump up ^ Cyrus: Persian conqueror of Babylon, 539 BC.
Jump up ^ Darius the Mede: No such person is known to history (see Levine, 2010, p. 1245, footnote 31). "Darius" is in any case a Persian, not a Median, name. The Persian army which captured Babylon was under the command of a certain Gobryas (or Gubaru), on behalf of Cyrus, the Persian king. The author of Daniel may have introduced the reference to a Mede in order to fulfill Isaiah and Jeremiah, who prophesied that the Medes would overthrow Babylon, confusing the events of 539 with those of 520 BC, when Darius I captured Babylon after an uprising. See Hammer, 1976, pp. 65–66.
Jump up ^ First year of Belshazzar: Probably 553 BC, when Belshazzar was given royal power by his father, Nabonidus. See Levine, 2010, p. 1248, footnote 7.1–8.
Jump up ^ "Third year of Cyrus": 536 BC. The author has apparently counted back seventy years to the "third year of Jehoiakim," 606 BC, to round out Daniel's prophetic ministry. See Towner, p. 149.
Jump up ^ "Son of man" (bar 'enaš in Hebrew) simply means "a human being", but in the context of Daniel 7 it may be a heavenly figure, possibly the archangel Michael functioning as a representative of the Jewish people (Collins 1977:144–46; opposed by Davies 1985:105–106). Scholars almost universally agree that this human figure represents "the people of the holy ones of the Most High" of Daniel 7:27, originally the maskilim community or group responsible for the composition of Daniel, but in later interpretation it is taken to mean the Jewish people as a whole. See Grabbe 2002a.
References[edit]
Citations[edit]
Jump up ^ Collins1984, p. 33.
Jump up ^ Reid 2000, p. 315.
^ Jump up to: a b Brettler 2005, p. 218.
Jump up ^ Bandstra 2008, p. 445.
^ Jump up to: a b Collins 2002, p. 2.
Jump up ^ Cross & Livingstone 2005, p. 452.
^ Jump up to: a b Towner 1984, pp. 2–3.
^ Jump up to: a b Collins 1984, p. 28.
Jump up ^ Provan 2003, p. 665.
^ Jump up to: a b Collins 1984, pp. 30–31.
Jump up ^ Collins 1984, p. 31.
Jump up ^ Redditt 2009, p. 177.
Jump up ^ Seow 2003, pp. 19–20.
Jump up ^ Seow 2003, pp. 31–33.
Jump up ^ Seow 2003, pp. 50–51.
Jump up ^ Levine 2010, p. 1241.
Jump up ^ Hammer 1976, pp. 57–60.
Jump up ^ Levine 2010, pp. 1245–1247.
Jump up ^ Levine 2010, pp. 1248–1249.
Jump up ^ Levine 2010, pp. 1249–1251.
Jump up ^ Levine 2010, pp. 1251–1252.
Jump up ^ McDonald 2012, p. 57.
Jump up ^ Harrington 1999, pp. 109–110.
Jump up ^ Grabbe 2010, pp. 6–13.
^ Jump up to: a b c Grabbe 2010, pp. 13–16.
Jump up ^ Sacchi 2010, pp. 225–226.
Jump up ^ Davies 2006, p. 407.
Jump up ^ Seow 2003, pp. 6–7.
Jump up ^ Seow 2003, p. 7.
Jump up ^ Collins 1993, p. 42.
Jump up ^ Collins 1984, p. 34.
^ Jump up to: a b Redditt 2008, pp. 176–177.
Jump up ^ Hammer 1976, p. 2.
Jump up ^ Wesselius 2002, p. 295.
Jump up ^ Grabbe 2002b, pp. 229–230, 243.
Jump up ^ Davies 2006, p. 340.
Jump up ^ Redditt 2008, p. 180.
Jump up ^ Collins 2003, p. 69.
Jump up ^ Seow 2003, p. 4.
Jump up ^ Collins 1984, p. 101.
Jump up ^ Hammer 1976, pp. 1–2.
Jump up ^ Harrington 1999, pp. 119–120.
Jump up ^ Spencer 2002, p. 89.
Jump up ^ Seow 2003, p. 3.
Jump up ^ VanderKam & Flint 2013, pp. 137–138.
Jump up ^ Crawford 2000, p. 73.
^ Jump up to: a b Davies 2006, pp. 397–406.
Jump up ^ Carroll 2000, pp. 420–421.
Jump up ^ Redditt 2009, p. 187.
Jump up ^ Seow 2003, pp. 1–2.
^ Jump up to: a b Niskanen 2004, pp. 27, 31.
Jump up ^ Towner 1984, pp. 34–36.
Jump up ^ Collins 1984, p. 80.
Jump up ^ Matthews & Moyes 2012, pp. 260,269.
Jump up ^ Seow 2003, pp. 3–4.
Jump up ^ Grabbe 2002a, pp. 60–61, 282.
Jump up ^ Collins 1984, p. 87.
Jump up ^ Collins 1998, pp. 108–109.
Jump up ^ Matthews & Moyer 2012, p. 260.
Jump up ^ Collins 1998, pp. 110–111.
Jump up ^ Carroll 2000, p. 420.
Jump up ^ Collins 1998, p. 114.
Jump up ^ Collins 1998, p. 99.
Jump up ^ Cohn 2002, pp. 86–87.
Jump up ^ Schwartz 1992, p. 2.
Jump up ^ Grabbe 2002, p. 244.
Jump up ^ Weber 2007, p. 374.
Jump up ^ Boyer 1992, pp. 24, 30–31.
^ Jump up to: a b Doukhan 2000, p. 9.
Jump up ^ Godden 2013, p. 231.
Bibliography[edit]
Bandstra, Barry L. (2008). Reading the Old Testament: An Introduction to the Hebrew Bible. Wadsworth Publishing Company. ISBN 0495391050.
Bar, Shaul (2001). A letter that has not been read: dreams in the Hebrew Bible. Cincinnati: Hebrew Union College Press. ISBN 978-0-87820-424-3.
Boyer, Paul S. (1992). When Time Shall Be No More: Prophecy Belief in Modern American Culture. Harvard University Press. ISBN 0-674-95129-8.
Brettler, Mark Zvi (2005). How To Read the Bible. Jewish Publication Society. ISBN 9780827610019.
Carroll, John T. (2000). "Eschatology". In Freedman, David Noel; Myers, Allen C. Eerdmans Dictionary of the Bible. Eerdmans. ISBN 9789053565032.
Cohn, Shaye J.D. (2006). From the Maccabees to the Mishnah. Westminster John Knox Press. ISBN 9780664227432.
Collins, John J. (1984). Daniel: With an Introduction to Apocalyptic Literature. Eerdmans. ISBN 9780802800206.
Collins, John J. (1993). Daniel. Fortress. ISBN 9780800660406.
Collins, John J. (1998). The Apocalyptic Imagination: An Introduction to Jewish Apocalyptic Literature. Eerdmans. ISBN 9780802843715. * Collins, John J. (2001). Seers, Sibyls, and Sages in Hellenistic-Roman Judaism. BRILL. ISBN 9780391041103.
Collins, John J. (2002). "Current Issues in the Study of Daniel". In Collins, John J.; Flint, Peter W.; VanEpps, Cameron. The Book of Daniel: Composition and Reception. BRILL. ISBN 9004116753.
Collins, John J. (2003). "From Prophecy to Apocalypticism: The Expectation of the End". In McGinn, Bernard; Collins, John J.; Stein, Stephen J. The Continuum History of Apocalypticism. Continuum. ISBN 9780826415202.
Collins, John J. (2013). "Daniel". In Lieb, Michael; Mason, Emma; Roberts, Jonathan. The Oxford Handbook of the Reception History of the Bible. Oxford UNiversity Press. ISBN 9780191649189.
Crawford, Sidnie White (2000). "Apocalyptic". In Freedman, David Noel; Myers, Allen C. Eerdmans Dictionary of the Bible. Eerdmans. ISBN 9789053565032.
Cross, Frank Leslie; Livingstone, Elizabeth A. (2005). The Oxford Dictionary of the Christian Church. Oxford University Press. ISBN 9780192802903.
Davies, Philip (2006). "Apocalyptic". In Rogerson, J. W.; Lieu, Judith M. The Oxford Handbook of Biblical Studies. Oxford Handbooks Online. ISBN 9780199254255.
DeChant, Dell (2009). "Apocalyptic Communities". In Neusner, Jacob. World Religions in America: An Introduction. Westminster John Knox Press. ISBN 9781611640472.
Doukhan, Jacques (2000). Secrets of Daniel: wisdom and dreams of a Jewish prince in exile. Review and Herald Pub Assoc. ISBN 9780828014243.
Dunn, James D.G. (2002). "The Danilic Son of Man in the New Testament". In Collins, John J.; Flint, Peter W.; VanEpps, Cameron. The Book of Daniel: Composition and Reception. BRILL. ISBN 0391041282.
Godden, Malcolm (2013). "Biblical Literature" The Old Testament". In Godden and, Malcolm; Lapidge, Michael. The Cambridge Companion to Old English Literature. Cambridge University Press. ISBN 9781107469211.
Grabbe, Lester L. (2002a). Judaic Religion in the Second Temple Period: Belief and Practice from the Exile to Yavneh. Routledge. ISBN 9780203461013.
Grabbe, Lester L. (2002b). "A Dan(iel) For All Seasons". In Collins, John J.; Flint, Peter W.; VanEpps, Cameron. The Book of Daniel: Composition and Reception. BRILL. ISBN 9004116753.
Grabbe, Lester L. (2010). An Introduction to Second Temple Judaism: History and Religion of the Jews in the Time of Nehemiah, the Maccabees, Hillel, and Jesus. Continuum. ISBN 9780567552488.
Hammer, Raymond (1976). The Book of Daniel. Cambridge University Press. ISBN 9780521097659.
Harrington, Daniel J. (1999). Invitation to the Apocrypha. Eerdmans. ISBN 9780802846334.
Hill, Andrew E. (2009). "Daniel". In Garland, David E.; Longman, Tremper. Daniel—Malachi. Zondervan. ISBN 9780310590545.
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Horsley, Richard A. (2007). Scribes, Visionaries, and the Politics of Second Temple Judea. Presbyterian Publishing Corp. ISBN 9780664229917.
Knibb, Michael (2002). "The Book of Daniel in its Context". In Collins, John J.; Flint, Peter W.; VanEpps, Cameron. The Book of Daniel: Composition and Reception. BRILL. ISBN 9004116753.
Levine, Amy-Jill (2010). "Daniel". In Coogan, Michael D.; Brettler, Marc Z.; Newsom, Carol A. The new Oxford annotated Bible with the Apocryphal/Deuterocanonical books : New Revised Standard Version. Oxford University Press. ISBN 9780199370504.
Lucas, Ernest C. (2005). "Daniel, Book of". In Vanhoozer, Kevin J.; Bartholomew, Craig G.; Treier, Daniel J. Dictionary for Theological Interpretation of the Bible. Baker Academic. ISBN 9780801026942.
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Provan, Iain (2003). "Daniel". In Dunn, James D.G.; Rogerson, John William. Eerdmans Commentary on the Bible. Eerdmans. ISBN 978-0-8028-3711-0.
Redditt, Paul L. (2009). Introduction to the Prophets. Eerdmans. ISBN 9780802828965.
Reid, Stephen Breck (2000). "Daniel, Book of". In Freedman, David Noel; Myers, Allen C. Eerdmans Dictionary of the Bible. Eerdmans. ISBN 9789053565032.
Rowland, Christopher (2007). "Apocalyptic Literature". In Hass, Andrew; Jasper, David; Jay, Elisabeth. The Oxford Handbook of English Literature and Theology. Oxford University Press. ISBN 9780199271979.
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Schwartz, Daniel R. (1992). Studies in the Jewish Background of Christianity. Mohr Siebeck. ISBN 9783161457982.
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Spencer, Richard A. (2002). "Additions to Daniel". In Mills, Watson E.; Wilson, Richard F. The Deuterocanonicals/Apocrypha. Mercer University Press. ISBN 9780865545106.
Towner, W. Sibley (1984). Daniel. Westminster John Knox Press. ISBN 9780664237561.
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VanderKam, James C.; Flint, Peter (2013). The meaning of the Dead Sea scrolls: their significance for understanding the Bible, Judaism, Jesus, and Christianity. HarperCollins. ISBN 9780062243300.
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External links[edit]
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The Book of Daniel * (Full text from St-Takla.org, also available in Arabic)
Daniel: 2013 Critical Translation with Audio Drama at biblicalaudio
Daniel an introduction
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Jewish Encyclopedia: Daniel
Daniel: Wise Man and Visionary, by Elias Bickerman
Aramaic of Daniel, Patrick Henry College
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text:
Special right triangle
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"90-45-45 triangle" redirects here. For the drawing tool, see 90-45-45 set square.
"30-60-90 triangle" redirects here. For the drawing tool, see 30-60-90 set square.
Position of some special triangles in an Euler diagram of types of triangles, using the definition that isosceles triangles have at least two equal sides, i.e. equilateral triangles are isosceles.
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.
Contents
[hide]
1 Angle-based
1.1 45°–45°–90° triangle
1.2 30°–60°–90° triangle
2 Side-based
2.1 Common Pythagorean triples
2.2 Almost-isosceles Pythagorean triples
2.3 Arithmetic and geometric progressions
2.4 Sides of regular polygons
3 See also
4 References
5 External links
Angle-based[edit]
Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering trigonometric functions of multiples of 30 and 45 degrees.
"Angle-based" special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or π/2 radians, is equal to the sum of the other two angles.
The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°.
Special triangles are used to aid in calculating common trigonometric functions, as below:
degrees
radians
gons
turns
sin
cos
tan
cotan
0° 0 0g 0 √0/2 = 0 √4/2 = 1 0 undefined
30° π/6 33 1/3g 1/12 √1/2 = 1/2 √3/2 1/√3 √3
45° π/4 50g 1/8 √2/2 = 1/√2 √2/2 = 1/√2 1 1
60° π/3 66 2/3g 1/6 √3/2 √1/2 = 1/2 √3 1/√3
90° π/2 100g 1/4 √4/2 = 1 √0/2 = 0 undefined 0
45°–45°–90°
30°–60°–90°
The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the equilateral/equiangular (60°–60°–60°) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides; see Triangle group.
45°–45°–90° triangle[edit]
The side lengths of a 45°–45°–90° triangle
In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or π radians. Hence, the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the Pythagorean theorem.
Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely √2/2.[1]:p.282,p.358
Of all right triangles, the 45°–45°–90° degree triangle has the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4.[1]:p.282
Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles.
30°–60°–90° triangle[edit]
The side lengths of a 30°–60°–90° triangle
This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° (π/6), 60° (π/3), and 90° (π/2). The sides are in the ratio 1 : √3 : 2.
The proof of this fact is clear using trigonometry. The geometric proof is:
Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1.
The fact that the remaining leg AD has length √3 follows immediately from the Pythagorean theorem.
The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α + δ, α + 2δ are the angles in the progression then the sum of the angles 3α + 3δ = 180°. After dividing by 3, the angle α + δ must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.
Side-based[edit]
Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that cannot all be rational numbers of degrees.[2] (This follows from Niven's theorem.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio
m2 − n2 : 2mn : m2 + n2
where m and n are any positive integers such that m > n.
Common Pythagorean triples[edit]
Main article: Pythagorean triple
There are several Pythagorean triples which are well-known, including those with sides in the ratios:
3:
4
:5
5:
12
:13
8:
15
:17
7:
24
:25
9:
40
:41
The 3 : 4 : 5 triangles are the only right triangles with edges in arithmetic progression. Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides.
The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras theoreæm was known at that time, has been much debated.[3] It was first conjectured by the historian Moritz Cantor in 1882.[3] It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement;[3] that Plutarch recorded in Isis and Osiris (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle;[3] and that the Berlin Papyrus 6619 from the Middle Kingdom of Egypt (before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other."[4] The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem."[3] Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles".[3]
The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones in lowest form in the list above) with both non-hypotenuse sides less than 256:
11:
60
:61
12:
35
:37
13:
84
:85
15:
112
:113
16:
63
:65
17:
144
:145
19:
180
:181
20:
21
:29
20:
99
:101
21:
220
:221
24:
143
:145
28:
45
:53
28:
195
:197
32:
255
:257
33:
56
:65
36:
77
:85
39:
80
:89
44:
117
:125
48:
55
:73
51:
140
:49
52:
165
:173
57:
176
:185
60:
91
:109
60:
221
:229
65:
72
:97
84:
187
:205
85:
132
:157
88:
105
:137
95:
168
:193
96:
247
:265
104:
153
:185
105:
208
:233
115:
252
:277
119:
120
:169
120:
209
:241
133:
156
:205
140:
171
:221
160:
231
:281
161:
240
:289
204:
253
:325
207:
224
:305
Almost-isosceles Pythagorean triples[edit]
Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is √2, but √2 cannot be expressed as a ratio of two integers. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the non-hypotenuse edges differ by one.[5][6] Such almost-isosceles right-angled triangles can be obtained recursively,
a0 = 1, b0 = 2
an = 2bn−1 + an−1
bn = 2an + bn−1
an is length of hypotenuse, n = 1, 2, 3, .... Equivalently,
( x − 1 2 ) 2 + ( x + 1 2 ) 2 = y 2 {\displaystyle ({\tfrac {x-1}{2}})^{2}+({\tfrac {x+1}{2}})^{2}=y^{2}}
where {x, y} are the solutions to the Pell equation x2 − 2y2 = −1, with the hypotenuse y being the odd terms of the Pell numbers 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... (sequence A000129 in the OEIS).. The smallest Pythagorean triples resulting are:[7]
3 :
4
: 5
20 :
21
: 29
119 :
120
: 169
696 :
697
: 985
4,059 :
4,060
: 5,741
23,660 :
23,661
: 33,461
137,903 :
137,904
: 195,025
803,760 :
803,761
: 1,136,689
4,684,659 :
4,684,660
: 6,625,109
Alternatively, the same triangles can be derived from the square triangular numbers.[8]
Arithmetic and geometric progressions[edit]
A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio.
Main article: Kepler triangle
The Kepler triangle is a right triangle whose sides are in a geometric progression. If the sides are formed from the geometric progression a, ar, ar2 then its common ratio r is given by r = √φ where φ is the golden ratio. Its sides are therefore in the ratio 1 : √φ : φ. Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in a geometric progression.
The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in an arithmetic progression.[9]
Sides of regular polygons[edit]
The sides of a pentagon, hexagon, and decagon, inscribed in congruent circles, form a right triangle
Let a = 2 sin π/10 = −1 + √5/2 = 1/φ be the side length of a regular decagon inscribed in the unit circle, where φ is the golden ratio. Let b = 2 sin π/6 = 1 be the side length of a regular hexagon in the unit circle, and let c = 2 sin π/5 = 5 − 5 2 {\displaystyle {\sqrt {\tfrac {5-{\sqrt {5}}}{2}}}} be the side length of a regular pentagon in the unit circle. Then a2 + b2 = c2, so these three lengths form the sides of a right triangle.[10] The same triangle forms half of a golden rectangle. It may also be found within a regular icosahedron of side length c: the shortest line segment from any vertex V to the plane of its five neighbors has length a, and the endpoints of this line segment together with any of the neighbors of V form the vertices of a right triangle with sides a, b, and c.[11]
See also[edit]
Integer triangle
Spiral of Theodorus
References[edit]
^ Jump up to: a b Posamentier, Alfred S., and Lehman, Ingmar. The Secrets of Triangles. Prometheus Books, 2012.
Jump up ^ Weisstein, Eric W. "Rational Triangle". MathWorld.
^ Jump up to: a b c d e f Cooke, Roger L. (2011). The History of Mathematics: A Brief Course (2nd ed.). John Wiley & Sons. pp. 237–238. ISBN 978-1-118-03024-0.
Jump up ^ Gillings, Richard J. (1982). Mathematics in the Time of the Pharaohs. Dover. p. 161.
Jump up ^ Forget, T. W.; Larkin, T. A. (1968), "Pythagorean triads of the form x, x + 1, z described by recurrence sequences" (PDF), Fibonacci Quarterly, 6 (3): 94–104 .
Jump up ^ Chen, C. C.; Peng, T. A. (1995), "Almost-isosceles right-angled triangles" (PDF), The Australasian Journal of Combinatorics, 11: 263–267, MR 1327342 .
Jump up ^ (sequence A001652 in the OEIS)
Jump up ^ Nyblom, M. A. (1998), "A note on the set of almost-isosceles right-angled triangles" (PDF), The Fibonacci Quarterly, 36 (4): 319–322, MR 1640364 .
Jump up ^ Beauregard, Raymond A.; Suryanarayan, E. R. (1997), "Arithmetic triangles", Mathematics Magazine, 70 (2): 105–115, MR 1448883, doi:10.2307/2691431 .
Jump up ^ Euclid's Elements, Book XIII, Proposition 10.
Jump up ^ nLab: pentagon decagon hexagon identity.
External links[edit]
3 : 4 : 5 triangle
30–60–90 triangle
45–45–90 triangle – with interactive animations
Retrieved from "https://en.wikipedia.org/w/index.php?title=Special_right_triangle&oldid=797001877"
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Tanya (name)
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Look up Tanya in Wiktionary, the free dictionary.
For other uses, see Tanya (disambiguation).
"LaTanya" redirects here. It is not to be confused with LaTonya or Latonia.
Tanya
Pronunciation
/ˈtɑːnjə/, /ˈtænjə/
Gender
Female
Language(s)
Aramaic
Ancient Greek
Latin
Russian
Sanskrit
Persian
Hebrew
Origin
Word/name
Slavic
Hebrew
Meaning
Short form of Tatiana
Ancient Greek (Establisher, Decider)
Latin ('tatius', great)
Russian (Ruler, Regent)
Sanskrit (Daughter)
Persian (unique girl)
Hebrew/Aramaic (it was taught in Biblical teaching)
Other names
Variant form(s)
LaTanya
Tanya is the Slavic hypocoristic of Tatiana. It is now quite commonly used as an independent given name in the English-speaking world.[1]
Tanya is also of Hebrew origin derived from the Aramaic term meaning 'it was taught in Baraita' which is Biblical teaching. Within Hasidic Judaism this refers to the Oral Torah. It is the first word used within the Tanya.[2]
Its popularity grew in many respects thanks to Alexander Pushkin's poem Eugene Onegin whose main character was Tatiana Larina, beloved by Onegin.
Variants include Tania (Ukrainian), Tanja (Bosnian/Croatian/Serbian, German, Danish, Estonian, Finnish, Dutch and Macedonian),[1] Tania (Romanian), Táňa (Czech) or Taanya (Levant and Indian subcontinent)[3]
It is the 237th most common name in the United States according to namestatistics.com, which uses US Census data.[4]
Contents
1 Tanya
2 Fictional
3 Latanya
4 References
5 See also
Tanya[edit]
Tanya Burr, English vlogger and blogger
Tanya Byron, a British psychologist
Tanya Chan, a legislator in Hong Kong
Tanya Chua, a singer in Singapore
Tania Coleridge, British model/Girl in Father Figure music video
Tanya Donelly, singer/guitarist for Throwing Muses, The Breeders, and Belly
Tanya Dubnicoff, Canadian track cyclist
Tanya Dziahileva, Belarusian model
Tanya Haden, daughter of jazz double bassist Charlie Haden, wife of actor Jack Black
Tanya Hansen, Norwegian pornographic actress.
Tanya Hughes, American high jumper
Tanya Hunks, Canadian swimmer
Tanya Oxley, Barbadian track and field sprinter
Tanya Mercado (as known as Gina Lynn), Puerto Rican pornographic actress
Tanya Plibersek, Australian politician and Minister for Health
Tanya Roberts, actress
Tanya Savicheva, Russian child diarist who died in 1944 during the Siege of Leningrad
Tanya Tucker, American country music singer
Fictional[edit]
Tanya Branning, in the soap opera Eastenders
Tanya Turner, the lead character on the ITV drama Footballers Wives
Tanya Mousekewitz, from the film An American Tail
Agent Tanya Adams, a character and unit type in the Command & Conquer: Red Alert video game series
Tanya Degurechaff, the eponymous young girl turned sadistic battalion commander in the light novel series, manga, and anime, Saga of Tanya the Evil
Tanya Sloan, a character from Power Rangers Turbo
Latanya[edit]
LaTanya Garrett U.S. politician
LaTanya Richardson (born 1949) U.S. actress
LaTanya Sheffield (born 1963) U.S. athlete
References[edit]
^ Jump up to: a b A Dictionary of First Names, Patrick Hanks & Flavia Hodges, Oxford University Press, 1990, ISBN 0-19-211651-7.
Jump up ^ Lehat, Sarah (2012-12-03). "Illustrated Tanya: The Iconography of a Classic Jewish Mystical Text (PHOTOS)". Huffington Post. Retrieved 2017-06-27.
Jump up ^ Samek, Ondřej Malačka, Jan. "Jméno | Táňa: 1731 | KdeJsme.cz | Četnost příjmení nebo jména v České republice". www.kdejsme.cz (in Czech). Retrieved 2017-06-27.
Jump up ^ "Tanya". namestatistics.com. Retrieved 9 October 2010.
See also[edit]
Tania (name)
Tanja (name)
Tonia (name)
Tonja (name)
Tonje (name)
Tonya (name)
Name list
This page or section lists people that share the same given name. If an internal link led you here, you may wish to change that link to point directly to the intended article.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Tanya_(name)&oldid=865657356"
Categories:
Given names
Russian feminine given names
Croatian feminine given names
Slovene feminine given names
Serbian feminine given names
Macedonian feminine given names
Bulgarian feminine given names
Slavic feminine given names
Arabic feminine given names
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CS1 Czech-language sources (cs)
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This page was last edited on 25 October 2018, at 10:03 (UTC).
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Tanya (disambiguation)

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Jack Black (actor)

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Tanya Mousekewitz

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Agent Tanya

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Saga of Tanya the Evil

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Tanya Sloan

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Tonja (name)

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Tanya – German
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Tanya – Icelandic
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