91-514: Euclid ( / ˈ j uː k l ɪ d / ; Ancient Greek : Εὐκλείδης ; fl. 300 BC) was an ancient Greek mathematician active as a geometer and logician . Considered the "father of geometry", he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry , involved innovations in combination with
182-473: A corruption of Greek mathematical terms. Euclid is best known for his thirteen-book treatise, the Elements ( Ancient Greek : Στοιχεῖα ; Stoicheia ), considered his magnum opus . Much of its content originates from earlier mathematicians, including Eudoxus , Hippocrates of Chios , Thales and Theaetetus , while other theorems are mentioned by Plato and Aristotle. It is difficult to differentiate
273-471: A modern axiomatization of the Elements . Ancient Greek language Ancient Greek ( Ἑλληνῐκή , Hellēnikḗ ; [hellɛːnikɛ́ː] ) includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek ( c. 1400–1200 BC ), Dark Ages ( c. 1200–800 BC ),
364-537: A pitch accent . In Modern Greek, all vowels and consonants are short. Many vowels and diphthongs once pronounced distinctly are pronounced as /i/ ( iotacism ). Some of the stops and glides in diphthongs have become fricatives , and the pitch accent has changed to a stress accent . Many of the changes took place in the Koine Greek period. The writing system of Modern Greek, however, does not reflect all pronunciation changes. The examples below represent Attic Greek in
455-509: A tomb was constructed for them. Shortly after this event, Ptolemy openly joined the coalition against Perdiccas. Perdiccas appears to have suspected Ptolemy of aiming for the throne himself, and may have decided that Ptolemy was his most dangerous rival. Ptolemy executed Cleomenes for spying on behalf of Perdiccas; this removed the chief check on his authority, and allowed Ptolemy to obtain the huge sum that Cleomenes had accumulated. In 321 BC, Perdiccas attempted to invade Egypt, only to fall at
546-773: A 'rebel in Cyprus' who was put to death by his half-brother Ptolemy II –as well as the daughters Ptolemais, who married Demetrius I of Macedon , and Lysandra , first married to Alexander V of Macedon and after to Lysimachus' son Agathocles . Ptolemy married once more to Berenice , Eurydice's cousin, who had come to Egypt as Eurydice's lady-in-waiting with the children from her first marriage to Philip . Their children were Arsinoe II , Philotera , and Ptolemy II. Their eldest child Arsinoe married Lysimachus, then her half-brother Ptolemy Keraunos, and finally her full brother Ptolemy II. In 285, Ptolemy made his son Ptolemy II his co-regent. His eldest legitimate son, Ptolemy Keraunos, fled to
637-704: A claim to Judea in southern Syria , which was disputed with the Seleucid king Seleucus I . He also took control of Cyprus and Cyrenaica , the latter of which was placed under the control of Ptolemy's stepson Magas . Ptolemy also commanded the construction of the Library of Alexandria and of the Lighthouse of Alexandria , one of the Seven Wonders of the Ancient World . Ptolemy I may have married Thaïs , his mistress during
728-434: A copy thereof, and is sometimes synonymous with 'geometry'. As with many ancient Greek mathematicians , the details of Euclid's life are mostly unknown. He is accepted as the author of four mostly extant treatises—the Elements , Optics , Data , Phaenomena —but besides this, there is nothing known for certain of him. The traditional narrative mainly follows the 5th century AD account by Proclus in his Commentary on
819-440: A distinctly unheroic role in proceedings to Perdiccas. More recently, J. Roisman has argued that the case for Ptolemy's blackening of Perdiccas and others has been much exaggerated. Ptolemy personally sponsored the great mathematician Euclid . He found Euclid's seminal work, the Elements , too difficult to study, so he asked if there were an easier way to master it. According to Proclus , Euclid famously quipped: "Sire, there
910-535: A fleet which detached the coastal towns of Phaselis , Xanthos , Kaunos , Iasos and Myndus in Lycia and Caria from Antigonus, then crossed into Greece, where he took possession of Corinth , Sicyon and Megara (308 BC). In 306, a great fleet under Demetrius attacked Cyprus, and Ptolemy's brother Menelaus was defeated and captured in another decisive Battle of Salamis . Ptolemy's complete loss of Cyprus followed. The satraps Antigonus and Demetrius now each assumed
1001-472: A great seat of Greek culture . Ptolemy I was the son of Arsinoe of Macedon by either her husband Lagus or Philip II of Macedon , the father of Alexander. However, the latter is unlikely and may be a myth fabricated to glorify the Ptolemaic Dynasty . Ptolemy was one of Alexander's most trusted companions and military officers. After the death of Alexander in 323 BC, Ptolemy retrieved his body as it
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#17328446414711092-472: A lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period. They have the same general outline but differ in some of the detail. The only attested dialect from this period is Mycenaean Greek , but its relationship to the historical dialects and
1183-419: A lesser degree. Pamphylian Greek , spoken in a small area on the southwestern coast of Anatolia and little preserved in inscriptions, may be either a fifth major dialect group, or it is Mycenaean Greek overlaid by Doric, with a non-Greek native influence. Regarding the speech of the ancient Macedonians diverse theories have been put forward, but the epigraphic activity and the archaeological discoveries in
1274-589: A mere conjecture. In any event, the contents of Euclid's work demonstrate familiarity with the Platonic geometry tradition. In his Collection , Pappus mentions that Apollonius studied with Euclid's students in Alexandria , and this has been taken to imply that Euclid worked and founded a mathematical tradition there. The city was founded by Alexander the Great in 331 BC, and the rule of Ptolemy I from 306 BC onwards gave it
1365-543: A prefix /e-/, called the augment . This was probably originally a separate word, meaning something like "then", added because tenses in PIE had primarily aspectual meaning. The augment is added to the indicative of the aorist, imperfect, and pluperfect, but not to any of the other forms of the aorist (no other forms of the imperfect and pluperfect exist). The two kinds of augment in Greek are syllabic and quantitative. The syllabic augment
1456-489: A stability which was relatively unique amid the chaotic wars over dividing Alexander's empire . Ptolemy began a process of hellenization and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education. Euclid is speculated to have been among the Musaeum's first scholars. Euclid's date of death is unknown; it has been speculated that he died c. 270 BC . Euclid
1547-402: A synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus , Hippocrates of Chios , Thales and Theaetetus . With Archimedes and Apollonius of Perga , Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics . Very little is known of Euclid's life, and most information comes from
1638-510: A vowel or /n s r/ ; final stops were lost, as in γάλα "milk", compared with γάλακτος "of milk" (genitive). Ancient Greek of the classical period also differed in both the inventory and distribution of original PIE phonemes due to numerous sound changes, notably the following: The pronunciation of Ancient Greek was very different from that of Modern Greek . Ancient Greek had long and short vowels ; many diphthongs ; double and single consonants; voiced, voiceless, and aspirated stops ; and
1729-487: A wider sense. There were several regional dialects of Ancient Greek; Attic Greek developed into Koine. Ancient Greek was a pluricentric language , divided into many dialects. The main dialect groups are Attic and Ionic , Aeolic , Arcadocypriot , and Doric , many of them with several subdivisions. Some dialects are found in standardized literary forms in literature , while others are attested only in inscriptions. There are also several historical forms. Homeric Greek
1820-556: Is a literary form of Archaic Greek (derived primarily from Ionic and Aeolic) used in the epic poems , the Iliad and the Odyssey , and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic and other Classical-era dialects. The origins, early form and development of the Hellenic language family are not well understood because of
1911-418: Is added to stems beginning with consonants, and simply prefixes e (stems beginning with r , however, add er ). The quantitative augment is added to stems beginning with vowels, and involves lengthening the vowel: Some verbs augment irregularly; the most common variation is e → ei . The irregularity can be explained diachronically by the loss of s between vowels, or that of the letter w , which affected
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#17328446414712002-644: Is called 'East Greek'. Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian Greek had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, as exemplified in the poems of the Boeotian poet Pindar who wrote in Doric with a small Aeolic admixture. Thessalian likewise had come under Northwest Greek influence, though to
2093-448: Is considered by some linguists to have been closely related to Greek . Among Indo-European branches with living descendants, Greek is often argued to have the closest genetic ties with Armenian (see also Graeco-Armenian ) and Indo-Iranian languages (see Graeco-Aryan ). Ancient Greek differs from Proto-Indo-European (PIE) and other Indo-European languages in certain ways. In phonotactics , ancient Greek words could end only in
2184-614: Is in Apollonius' prefatory letter to the Conics (early 2nd century BC): "The third book of the Conics contains many astonishing theorems that are useful for both the syntheses and the determinations of number of solutions of solid loci . Most of these, and the finest of them, are novel. And when we discovered them we realized that Euclid had not made the synthesis of the locus on three and four lines but only an accidental fragment of it, and even that
2275-461: Is no royal road to geometry". This anecdote is questionable since a very similar interaction between Menaechmus and Alexander the Great is recorded from Stobaeus . Both accounts were written in the 5th century AD, neither indicates its source, and neither appears in ancient Greek literature. Any firm dating of Euclid's activity c. 300 BC is called into question by a lack of contemporary references. The earliest original reference to Euclid
2366-416: Is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Euclid of Megara , a pupil of Socrates included in dialogues of Plato with whom he was historically conflated. Valerius Maximus , the 1st century AD Roman compiler of anecdotes, mistakenly substituted Euclid's name for Eudoxus (4th century BC) as the mathematician to whom Plato sent those asking how to double
2457-583: Is some speculation that Euclid studied at the Platonic Academy and later taught at the Musaeum ; he is regarded as bridging the earlier Platonic tradition in Athens with the later tradition of Alexandria. In the Elements , Euclid deduced the theorems from a small set of axioms . He also wrote works on perspective , conic sections , spherical geometry , number theory , and mathematical rigour . In addition to
2548-534: Is unknown if Euclid intended the Elements as a textbook, but its method of presentation makes it a natural fit. As a whole, the authorial voice remains general and impersonal. Book 1 of the Elements is foundational for the entire text. It begins with a series of 20 definitions for basic geometric concepts such as lines , angles and various regular polygons . Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions. These assumptions are intended to provide
2639-723: Is unknown; some scholars estimate around 330 or 325 BC, but others refrain from speculating. It is presumed that he was of Greek descent, but his birthplace is unknown. Proclus held that Euclid followed the Platonic tradition , but there is no definitive confirmation for this. It is unlikely he was a contemporary of Plato, so it is often presumed that he was educated by Plato's disciples at the Platonic Academy in Athens. Historian Thomas Heath supported this theory, noting that most capable geometers lived in Athens, including many of those whose work Euclid built on; historian Michalis Sialaros considers this
2730-658: The Archaic or Epic period ( c. 800–500 BC ), and the Classical period ( c. 500–300 BC ). Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers . It has contributed many words to English vocabulary and has been a standard subject of study in educational institutions of the Western world since the Renaissance . This article primarily contains information about
2821-475: The Elements in works whose dates are firmly known are not until the 2nd century AD, by Galen and Alexander of Aphrodisias ; by this time it was a standard school text. Some ancient Greek mathematicians mention Euclid by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author of Elements "). In the Middle Ages, some scholars contended Euclid was not a historical personage and that his name arose from
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2912-412: The Elements reveals authorial control beyond the limits of a mere editor". The Elements does not exclusively discuss geometry as is sometimes believed. It is traditionally divided into three topics: plane geometry (books 1–6), basic number theory (books 7–10) and solid geometry (books 11–13)—though book 5 (on proportions) and 10 (on irrational lines) do not exactly fit this scheme. The heart of
3003-598: The Elements was published in 1570 by Henry Billingsley and John Dee . The mathematician Oliver Byrne published a well-known version of the Elements in 1847 entitled The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners , which included colored diagrams intended to increase its pedagogical effect. David Hilbert authored
3094-647: The Elements , Euclid wrote a central early text in the optics field, Optics , and lesser-known works including Data and Phaenomena . Euclid's authorship of On Divisions of Figures and Catoptrics has been questioned. He is thought to have written many lost works . The English name 'Euclid' is the anglicized version of the Ancient Greek name Eukleídes ( Εὐκλείδης ). It is derived from ' eu- ' ( εὖ ; 'well') and 'klês' ( -κλῆς ; 'fame'), meaning "renowned, glorious". In English, by metonymy , 'Euclid' can mean his most well-known work, Euclid's Elements , or
3185-410: The Elements , at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements , with definitions and proved propositions. Four other works are credibly attributed to Euclid, but have been lost. Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity. Many commentators cite him as one of
3276-539: The Elements , book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes. The final three books (11–13) primarily discuss solid geometry . By introducing a list of 37 definitions, Book 11 contextualizes the next two. Although its foundational character resembles Book 1, unlike the latter it features no axiomatic system or postulates. The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and parallelepipedal solids (24–37). In addition to
3367-537: The Epic and Classical periods of the language, which are the best-attested periods and considered most typical of Ancient Greek. From the Hellenistic period ( c. 300 BC ), Ancient Greek was followed by Koine Greek , which is regarded as a separate historical stage, though its earliest form closely resembles Attic Greek , and its latest form approaches Medieval Greek , and Koine may be classified as Ancient Greek in
3458-698: The Greek region of Macedonia during the last decades has brought to light documents, among which the first texts written in Macedonian , such as the Pella curse tablet , as Hatzopoulos and other scholars note. Based on the conclusions drawn by several studies and findings such as Pella curse tablet , Emilio Crespo and other scholars suggest that ancient Macedonian was a Northwest Doric dialect , which shares isoglosses with its neighboring Thessalian dialects spoken in northeastern Thessaly . Some have also suggested an Aeolic Greek classification. The Lesbian dialect
3549-599: The Seleucid and Ptolemaic dynasties. Henceforth, Ptolemy seems to have involved himself as little as possible in the rivalries between Asia Minor and Greece ; he lost what he held in Greece, but reconquered Cyprus in 295/294. Cyrenaica , after a series of rebellions, was finally subjugated in about 300 and placed under his stepson Magas . While Alexander was alive, Ptolemy had three children with his mistress Thaïs , who may also have been his wife: Lagus; Leontiscus; and Eirene, who
3640-538: The area of triangles and parallelograms (35–45); and the Pythagorean theorem (46–48). The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate". Book 2 is traditionally understood as concerning " geometric algebra ", though this interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since
3731-494: The pentagon . Book 5 is among the work's most important sections and presents what is usually termed as the "general theory of proportion". Book 6 utilizes the "theory of ratios " in the context of plane geometry. It is built almost entirely of its first proposition: "Triangles and parallelograms which are under the same height are to one another as their bases". From Book 7 onwards, the mathematician Benno Artmann [ de ] notes that "Euclid starts afresh. Nothing from
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3822-501: The present , future , and imperfect are imperfective in aspect; the aorist , present perfect , pluperfect and future perfect are perfective in aspect. Most tenses display all four moods and three voices, although there is no future subjunctive or imperative. Also, there is no imperfect subjunctive, optative or imperative. The infinitives and participles correspond to the finite combinations of tense, aspect, and voice. The indicative of past tenses adds (conceptually, at least)
3913-1031: The 5th century BC. Ancient pronunciation cannot be reconstructed with certainty, but Greek from the period is well documented, and there is little disagreement among linguists as to the general nature of the sounds that the letters represent. /oː/ raised to [uː] , probably by the 4th century BC. Greek, like all of the older Indo-European languages , is highly inflected. It is highly archaic in its preservation of Proto-Indo-European forms. In ancient Greek, nouns (including proper nouns) have five cases ( nominative , genitive , dative , accusative , and vocative ), three genders ( masculine , feminine , and neuter ), and three numbers (singular, dual , and plural ). Verbs have four moods ( indicative , imperative , subjunctive , and optative ) and three voices (active, middle, and passive ), as well as three persons (first, second, and third) and various other forms. Verbs are conjugated through seven combinations of tenses and aspect (generally simply called "tenses"):
4004-490: The Archaic period of ancient Greek (see Homeric Greek for more details): Μῆνιν ἄειδε, θεά, Πηληϊάδεω Ἀχιλῆος οὐλομένην, ἣ μυρί' Ἀχαιοῖς ἄλγε' ἔθηκε, πολλὰς δ' ἰφθίμους ψυχὰς Ἄϊδι προΐαψεν ἡρώων, αὐτοὺς δὲ ἑλώρια τεῦχε κύνεσσιν οἰωνοῖσί τε πᾶσι· Διὸς δ' ἐτελείετο βουλή· ἐξ οὗ δὴ τὰ πρῶτα διαστήτην ἐρίσαντε Ἀτρεΐδης τε ἄναξ ἀνδρῶν καὶ δῖος Ἀχιλλεύς. The beginning of Apology by Plato exemplifies Attic Greek from
4095-626: The Classical period of ancient Greek. (The second line is the IPA , the third is transliterated into the Latin alphabet using a modern version of the Erasmian scheme .) Ὅτι [hóti Hóti μὲν men mèn ὑμεῖς, hyːmêːs hūmeîs, Ptolemy I Soter Ptolemy I Soter ( / ˈ t ɒ l əm i / ; Greek : Πτολεμαῖος Σωτήρ , Ptolemaîos Sōtḗr , "Ptolemy the Savior "; c. 367 BC – January 282 BC)
4186-483: The First Book of Euclid's Elements , as well as a few anecdotes from Pappus of Alexandria in the early 4th century. According to Proclus, Euclid lived shortly after several of Plato 's ( d. 347 BC) followers and before the mathematician Archimedes ( c. 287 – c. 212 BC); specifically, Proclus placed Euclid during the rule of Ptolemy I ( r. 305/304–282 BC). Euclid's birthdate
4277-479: The Great Library of Alexandria . The Ptolemaic dynasty which he founded ruled Egypt for nearly three hundred years. It was a Hellenistic kingdom known for its capital Alexandria, which became a center of Greek culture . Ptolemaic rule ended with the death of Cleopatra VII in 30 BC. Ptolemy himself wrote an eyewitness history of Alexander's campaigns (now lost). In the second century AD, Ptolemy's history
4368-708: The Great. On his deathbed , Alexander wished to be buried at the Temple of Zeus Ammon in the Siwa Oasis of ancient Libya instead of the royal tombs of Aigai in Macedon. However, his successors including Perdiccas attempted to bury his body in Macedon instead. In late 322 or early 321 BC, Alexander's body was in Syria, on its way to Macedon, when it was captured by Ptolemy I. He brought Alexander's remains back to Egypt, interring them at Memphis , but they were later moved to Alexandria where
4459-550: The aorist. Following Homer 's practice, the augment is sometimes not made in poetry , especially epic poetry. The augment sometimes substitutes for reduplication; see below. Almost all forms of the perfect, pluperfect, and future perfect reduplicate the initial syllable of the verb stem. (A few irregular forms of perfect do not reduplicate, whereas a handful of irregular aorists reduplicate.) The three types of reduplication are: Irregular duplication can be understood diachronically. For example, lambanō (root lab ) has
4550-419: The augment when it was word-initial. In verbs with a preposition as a prefix, the augment is placed not at the start of the word, but between the preposition and the original verb. For example, προσ(-)βάλλω (I attack) goes to προσ έ βαλoν in the aorist. However compound verbs consisting of a prefix that is not a preposition retain the augment at the start of the word: αὐτο(-)μολῶ goes to ηὐ τομόλησα in
4641-540: The author "whom I chiefly follow", and in his Preface writes that Ptolemy seemed to him to be a particularly trustworthy source, "not only because he was present with Alexander on campaign, but also because he was himself a king, and hence lying would be more dishonourable for him than for anyone else". Ptolemy's lost history was long considered an objective work, distinguished by its straightforward honesty and sobriety, but more recent work has called this assessment into question. R. M. Errington argued that Ptolemy's history
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#17328446414714732-459: The campaign against the rebel Bessus whom his own guards captured and handed over to Ptolemy, who then handed him over to Alexander for execution. When Alexander died in 323 BC, Ptolemy is said to have instigated the settlement of the empire made at Babylon . Through the Partition of Babylon , he was appointed satrap of Egypt , under the nominal kings Philip III and the infant Alexander IV ;
4823-438: The center of Greek scholarship, this division of people and language is quite similar to the results of modern archaeological-linguistic investigation. One standard formulation for the dialects is: West vs. non-West Greek is the strongest-marked and earliest division, with non-West in subsets of Ionic-Attic (or Attic-Ionic) and Aeolic vs. Arcadocypriot, or Aeolic and Arcado-Cypriot vs. Ionic-Attic. Often non-West
4914-478: The court of Lysimachus. Ptolemy I died in January 282 aged 84 or 85. Shrewd and cautious, he had a compact and well-ordered realm to show at the end of forty years of war. His reputation for good nature and liberality attached the floating soldier-class of Macedonians and other Greeks to his service, and was not insignificant; nor did he wholly neglect conciliation of the natives. He was a ready patron of letters, founding
5005-437: The cube . Perhaps on the basis of this mention of a mathematical Euclid roughly a century early, Euclid became mixed up with Euclid of Megara in medieval Byzantine sources (now lost), eventually leading Euclid the mathematician to be ascribed details of both men's biographies and described as Megarensis ( lit. ' of Megara ' ). The Byzantine scholar Theodore Metochites ( c. 1300 ) explicitly conflated
5096-563: The dialect of Sparta ), and Northern Peloponnesus Doric (including Corinthian ). All the groups were represented by colonies beyond Greece proper as well, and these colonies generally developed local characteristics, often under the influence of settlers or neighbors speaking different Greek dialects. After the conquests of Alexander the Great in the late 4th century BC, a new international dialect known as Koine or Common Greek developed, largely based on Attic Greek , but with influence from other dialects. This dialect slowly replaced most of
5187-484: The fictionalization was done to strengthen the connection between a revered mathematician and the Arab world. There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man". The best known of these is Proclus' story about Ptolemy asking Euclid if there was a quicker path to learning geometry than reading his Elements , which Euclid replied with "there
5278-515: The former satrap, the Greek Cleomenes , stayed on as his deputy. Ptolemy quickly moved, without authorization, to subjugate Cyrenaica . By custom, kings in Macedonia asserted their right to the throne by burying their predecessor. Probably because he wanted to pre-empt Perdiccas , the imperial regent, from staking his claim in this way, Ptolemy took great pains in acquiring the body of Alexander
5369-407: The foundations of even nascent algebra occurred many centuries later. The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes. It focuses on the area of rectangles and squares (see Quadrature ), and leads up to a geometric precursor of the law of cosines . Book 3 focuses on circles, while the 4th discusses regular polygons , especially
5460-459: The fugitive satrap of Babylonia, both invaded Syria, and defeated Demetrius I , the son of Antigonus, in the Battle of Gaza . Again he occupied Syria, and again—after only a few months, when Demetrius had won a battle over his general, and Antigonus entered Syria in force—he evacuated it. In 311, a peace was concluded between the combatants. Soon after this, the surviving 13-year-old king, Alexander IV,
5551-514: The half-brother of Alexander the Great . It is probable that this is a later myth fabricated to glorify the Ptolemaic dynasty. However, through his mother Ptolemy may have been a great-grandson of Amyntas I of Macedon , making him a member of the Argead royal house and a distant relative of Alexander, who was a great-great-grandson of Amyntas. Ptolemy served with Alexander from his first campaigns, and
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#17328446414715642-538: The hands of his own men. Ptolemy's decision to defend the Nile against Perdiccas ended in fiasco for Perdiccas, with the loss of 2,000 men. This failure was a fatal blow to Perdiccas' reputation, and he was murdered in his tent by two of his subordinates. Ptolemy immediately crossed the Nile, to provide supplies to what had the day before been an enemy army. Ptolemy was offered the regency in place of Perdiccas, but he declined. Ptolemy
5733-556: The historical Dorians . The invasion is known to have displaced population to the later Attic-Ionic regions, who regarded themselves as descendants of the population displaced by or contending with the Dorians. The Greeks of this period believed there were three major divisions of all Greek people – Dorians, Aeolians, and Ionians (including Athenians), each with their own defining and distinctive dialects. Allowing for their oversight of Arcadian, an obscure mountain dialect, and Cypriot, far from
5824-472: The historical circumstances of the times imply that the overall groups already existed in some form. Scholars assume that major Ancient Greek period dialect groups developed not later than 1120 BC, at the time of the Dorian invasions —and that their first appearances as precise alphabetic writing began in the 8th century BC. The invasion would not be "Dorian" unless the invaders had some cultural relationship to
5915-534: The life of Alexander; he is known to have married the Persian noblewoman Artakama on Alexander's orders. He later married Eurydice , daughter of the Macedonian regent Antipater ; their sons Ptolemy Keraunos and Meleager ruled in turn as kings of Macedon. Ptolemy's final marriage was to Eurydice's cousin and lady-in-waiting, Berenice I . Upon his death, he was succeeded by his son with Berenice, Ptolemy II . Ptolemy
6006-486: The logical basis for every subsequent theorem, i.e. serve as an axiomatic system . The common notions exclusively concern the comparison of magnitudes . While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous. Book 1 also includes 48 propositions, which can be loosely divided into those concerning basic theorems and constructions of plane geometry and triangle congruence (1–26); parallel lines (27–34);
6097-589: The lunar crater Euclides , and the minor planet 4354 Euclides . The Elements is often considered after the Bible as the most frequently translated, published, and studied book in the Western World 's history. With Aristotle's Metaphysics , the Elements is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Arab and Latin worlds. The first English edition of
6188-464: The most influential figures in the history of mathematics . The geometrical system established by the Elements long dominated the field; however, today that system is often referred to as ' Euclidean geometry ' to distinguish it from other non-Euclidean geometries discovered in the early 19th century. Among Euclid's many namesakes are the European Space Agency 's (ESA) Euclid spacecraft,
6279-401: The news came that Antigonus had been defeated and slain by Lysimachus and Seleucus at the Battle of Ipsus in 301, he occupied Syria a fourth time. The other members of the coalition had assigned all Syria to Seleucus, after what they regarded as Ptolemy's desertion, and for the next hundred years, the question of the ownership of southern Syria (i.e., Judea) produced recurring warfare between
6370-499: The older dialects, although the Doric dialect has survived in the Tsakonian language , which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek . By about the 6th century AD, the Koine had slowly metamorphosed into Medieval Greek . Phrygian is an extinct Indo-European language of West and Central Anatolia , which
6461-487: The perfect stem eilēpha (not * lelēpha ) because it was originally slambanō , with perfect seslēpha , becoming eilēpha through compensatory lengthening. Reduplication is also visible in the present tense stems of certain verbs. These stems add a syllable consisting of the root's initial consonant followed by i . A nasal stop appears after the reduplication in some verbs. The earliest extant examples of ancient Greek writing ( c. 1450 BC ) are in
6552-524: The preceding books is used". Number theory is covered by books 7 to 10, the former beginning with a set of 22 definitions for parity , prime numbers and other arithmetic-related concepts. Book 7 includes the Euclidean algorithm , a method for finding the greatest common divisor of two numbers. The 8th book discusses geometric progressions , while book 9 includes the proposition, now called Euclid's theorem , that there are infinitely many prime numbers . Of
6643-403: The scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philosopher Euclid of Megara . It is now generally accepted that he spent his career in Alexandria and lived around 300 BC, after Plato 's students and before Archimedes. There
6734-517: The syllabic script Linear B . Beginning in the 8th century BC, however, the Greek alphabet became standard, albeit with some variation among dialects. Early texts are written in boustrophedon style, but left-to-right became standard during the classic period. Modern editions of ancient Greek texts are usually written with accents and breathing marks , interword spacing , modern punctuation , and sometimes mixed case , but these were all introduced later. The beginning of Homer 's Iliad exemplifies
6825-531: The text is the theorems scattered throughout. Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles". The first group includes statements labeled as a "definition" ( Ancient Greek : ὅρος or ὁρισμός ), "postulate" ( αἴτημα ), or a "common notion" ( κοινὴ ἔννοια ); only the first book includes postulates—later known as axioms —and common notions. The second group consists of propositions, presented alongside mathematical proofs and diagrams. It
6916-417: The title of king; Ptolemy, as well as Cassander , Lysimachus and Seleucus I Nicator , responded by doing the same. In the winter of 306 BC, Antigonus tried to follow up his victory in Cyprus by invading Egypt; but Ptolemy was strongest there, and successfully held the frontier against him. Ptolemy led no further overseas expeditions against Antigonus. However, he did send great assistance to Rhodes when it
7007-764: The two Euclids, as did printer Erhard Ratdolt 's 1482 editio princeps of Campanus of Novara 's Latin translation of the Elements . After the mathematician Bartolomeo Zamberti [ fr ; de ] appended most of the extant biographical fragments about either Euclid to the preface of his 1505 translation of the Elements , subsequent publications passed on this identification. Later Renaissance scholars, particularly Peter Ramus , reevaluated this claim, proving it false via issues in chronology and contradiction in early sources. Medieval Arabic sources give vast amounts of information concerning Euclid's life, but are completely unverifiable. Most scholars consider them of dubious authenticity; Heath in particular contends that
7098-494: The work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics. The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps" and the historian Serafina Cuomo described it as a "reservoir of results". Despite this, Sialaros furthers that "the remarkably tight structure of
7189-458: Was besieged by Demetrius (305/304) . The Rhodians granted divine honors to Ptolemy as a result of the lifting of the siege. When the coalition against Antigonus was renewed in 302, Ptolemy joined it, and invaded Syria a third time, while Antigonus was engaged with Lysimachus in Asia Minor . On hearing a report that Antigonus had won a decisive victory there, he once again evacuated Syria. But when
7280-475: Was Aeolic. For example, fragments of the works of the poet Sappho from the island of Lesbos are in Aeolian. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, or to an island. Doric notably had several intermediate divisions as well, into Island Doric (including Cretan Doric ), Southern Peloponnesus Doric (including Laconian ,
7371-468: Was a Macedonian Greek general, historian, and successor of Alexander the Great who went on to found the Ptolemaic Kingdom centered on Egypt . Ptolemy was basileus and pharaoh of Ptolemaic Egypt from 305/304 BC to his death in 282 BC, and his descendants continued to rule Egypt until 30 BC. During their rule, Egypt became a thriving bastion of Hellenistic civilization and Alexandria
7462-535: Was among the seven somatophylakes (bodyguards) of Alexander. He played a principal part in the later campaigns in Afghanistan and India . He participated in the Battle of Issus , commanding troops on the left wing under the authority of Parmenion. Later he accompanied Alexander during his journey to the Oracle in the Siwa Oasis where he was proclaimed a son of Zeus . Ptolemy had his first independent command during
7553-581: Was born in 367 BC in the ancient kingdom of Macedon . His mother was Arsinoe . According to Satyrus the Peripatetic , Arsinoe was a descendant of Alexander I of Macedon and thus a member of the Argead dynasty , claiming ultimate descent from Heracles . Ostensibly, Ptolemy's father was Lagus , a Macedonian nobleman from Eordaea , but many ancient sources claim that he was actually an illegitimate son of Philip II of Macedon . If true, this would have made Ptolemy
7644-547: Was characterised by persistent bias and self-aggrandisement, and by systematic blackening of the reputation of Perdiccas , one of Ptolemy's chief dynastic rivals after Alexander's death. For example, Arrian's account of the fall of Thebes in 335 BC ( Anabasis 1.8.1–1.8.8 , a rare section of narrative explicitly attributed to Ptolemy by Arrian) shows several significant variations from the parallel account preserved in Diodorus Siculus ( 17.11–12 ), most notably in attributing
7735-418: Was consistent in his policy of securing a power base, while never succumbing to the temptation of risking all to succeed Alexander. In the long wars that followed between the different Diadochi , Ptolemy's first goal was to hold Egypt securely, and his second was to secure control in the outlying areas: Cyrenaica and Cyprus , as well as Syria , including the province of Judea . His first occupation of Syria
7826-489: Was en route to be buried in Macedon, placing it in Memphis instead, where it was later moved to Alexandria in a new tomb . Afterwards he joined a coalition against Perdiccas , the royal regent over Philip III of Macedon . The latter invaded Egypt but was assassinated by his own officers in 320 BC, allowing Ptolemy I to consolidate his control over the country. After a series of wars between Alexander's successors , Ptolemy gained
7917-500: Was given in marriage to Eunostos of Soloi in Cyprus. During the Susa weddings , Ptolemy married Persian noblewoman Artakama , as ordered by Alexander the Great. Around 322 BC , he married Eurydice , daughter of Antipater , regent of Macedonia. They had five children before she was repudiated: three sons– Ptolemy Ceraunus , king of Macedon from 281 BC to 279 BC ; his brother and successor Meleager , who ruled for two months in 279 BC; and
8008-470: Was in 318, and he established at the same time a protectorate over the petty kings of Cyprus. When Antigonus I , master of Asia in 315, showed expansionist ambitions, Ptolemy joined the coalition against him, and on the outbreak of war, evacuated Syria. In Cyprus, he fought the partisans of Antigonus, and re-conquered the island (313). A revolt in Cyrene was crushed the same year. In 312, Ptolemy and Seleucus ,
8099-471: Was murdered in Macedonia on the orders of Cassander, leaving the satrap of Egypt absolutely his own master. The peace did not last long, early in 310 he was informed that his ally Nicocles of Paphos was planning to defect to Antigonus; he sent some agents, who together with his brother Menelaus , who was still on Cyprus with an army, dealt with the situation, they surrounded Nicocles palace and forced him to commit suicide. In 309 Ptolemy personally commanded
8190-553: Was not felicitously done." The Elements is speculated to have been at least partly in circulation by the 3rd century BC, as Archimedes and Apollonius take several of its propositions for granted; however, Archimedes employs an older variant of the theory of proportions than the one found in the Elements . The oldest physical copies of material included in the Elements , dating from roughly 100 AD, can be found on papyrus fragments unearthed in an ancient rubbish heap from Oxyrhynchus , Roman Egypt . The oldest extant direct citations to
8281-516: Was used by Arrian of Nicomedia as one of his two main primary sources (alongside the history of Aristobulus of Cassandreia ) for his own extant Anabasis of Alexander, and hence large parts of Ptolemy's history can be assumed to survive in paraphrase or précis in Arrian's work. Arrian cites Ptolemy by name on only a few occasions, but it is likely that large stretches of Arrian's Anabasis reflect Ptolemy's version of events. Arrian once names Ptolemy as
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