90-399: Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish mathematician , physicist and astronomer . He was Andrews Professor of Astronomy at Trinity College Dublin . Hamilton was Dunsink's third director, having worked there from 1827 to 1865. His career included the study of geometrical optics , Fourier analysis , and quaternions , the last of which made him one of
180-821: A Latin copy of Euclid ; and at twelve he studied Newton 's Arithmetica Universalis . By age 16, he had covered much of the Principia , as well as some more recent works on analytic geometry and differential calculus . In mid-1822, Hamilton began a systematic study of Laplace 's Mécanique Céleste . During this period, he encountered what he believed to be a logical error in Mécanique Céleste , an observation which led Hamilton to be introduced to John Brinkley , then Royal Astronomer of Ireland. In November and December of 1822, he completed his first three original mathematical papers. On his first visit to Dunsink Observatory, he showed two of them to Brinkley, who requested that
270-537: 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
360-458: A basic understanding of them. At the age of seven, he had already made progress in Hebrew , and before he was 13, he had acquired, under the care of his uncle, a dozen languages: classical and modern European languages, Persian , Arabic , Hindustani , Sanskrit , Marathi and Malay . The emphasis of Hamilton's early education on languages is attributed to the wish of his father to see him employed by
450-639: A corresponding member of the Saint Petersburg Academy of Sciences . Later, in 1864, the newly established United States National Academy of Sciences elected its first Foreign Associates, and decided to put Hamilton's name on top of their list. A plaque under the Broom Bridge, associated with the discovery of quaternions, was unveiled by Éamon de Valera on 13 November 1958. Since 1989, the National University of Ireland, Maynooth , has organised
540-465: A deeper mathematical structure than had been previously understood, in particular a symmetry between momentum and position. The credit for discovering what are now called the Lagrangian and Lagrange's equations also belongs to Hamilton. Both the Lagrangian mechanics and Hamiltonian approaches have proven important in the study of continuous classical systems in physics, and quantum mechanical systems:
630-471: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include
720-472: 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
810-419: 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
900-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while
990-771: A party with John Marshall and his family, he stayed at Dunsink with Hamilton. On a second tour in England with Nimmo in 1831, Hamilton parted from him at Birmingham , to visit the Lawrence sisters and family on his mother's side in the Liverpool area. They met up again in the Lake District , where they climbed Helvellyn and had tea with Wordsworth. Hamilton returned to Dublin, via Edinburgh and Glasgow. Hamilton visited Samuel Taylor Coleridge at Highgate , in 1832, helped by an unexpected letter of introduction given to him by Sarah Lawrence on
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#17329098568041080-597: A pilgrimage called the Hamilton Walk , in which mathematicians take a walk from Dunsink Observatory to the bridge, where no trace of the carving remains, though a stone plaque does commemorate the discovery. The Hamilton Institute is an applied mathematics research institute at Maynooth University and the Royal Irish Academy holds an annual public Hamilton lecture at which Murray Gell-Mann , Frank Wilczek , Andrew Wiles and Timothy Gowers have all spoken. 2005
1170-429: A poet. In 1825, Hamilton met Arabella Lawrence , younger sister of Sarah Lawrence , a significant correspondent and frank critic of his poetry. It was a contact he made through Maria Edgeworth's circle. Hamilton, now Royal Astronomer of Ireland, took up residence at Dunsink Observatory where he spent the rest of his life. He was there from 1827 until his death in 1865. In his early years at Dunsink, Hamilton observed
1260-766: A political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages
1350-543: 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
1440-483: A quaternion as an ordered four-element multiple of real numbers, and described the first element as the "scalar" part, and the remaining three as the "vector" part. He coined the neologisms "tensor" and "scalar", and was the first to use the word "vector" in the modern sense. Hamilton looked into the solution of the quintic in the theory of equations , examining the results arrived at by Niels Henrik Abel , George Jerrard and others in their respective research. There
1530-671: 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 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
1620-430: A single function, now known as Hamilton's principal function , that brings together mechanics and optical theory. It helped to establish the foundations of the wave theory of light in mathematical physics . He proposed it when he first predicted its existence in the third supplement to his Systems of Rays , read in 1832. The Royal Irish Academy paper was finally entitled Theory of Systems of Rays (23 April 1827), and
1710-609: A standard subject of study in educational institutions of the Western world since the Renaissance . This article primarily contains information about 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
1800-581: A visit to Liverpool in March of that year. He also paid a call, with Arabella, on the family of William Roscoe , who had died in 1831. He was a Christian, described as "a lover of the Bible, an orthodox and attached member of the Established Church", and as having a "profound conviction of the truth of revealed religion". While attending Trinity College, Hamilton proposed to his friend's sister, whose refusal drove
1890-510: 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
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#17329098568041980-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics
2070-516: Is Hamilton's paper on fluctuating functions in Fourier analysis , and the invention of the hodograph . Of his investigations into the solutions, especially by numerical approximation , of certain classes of physically-important differential equations, only parts were published, at intervals, in the Philosophical Magazine . Hamilton also introduced the icosian game or Hamilton's puzzle . It
2160-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics
2250-556: 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
2340-426: Is a prediction for transparent biaxial crystals (i.e. monoclinic , orthorhombic or triclinic crystals). A ray of light entering such a crystal at a certain angle would emerge as a hollow cone of rays. This discovery was known as conical refraction . Hamilton found it from the geometry of the wave surface introduced by Augustin-Jean Fresnel , which has singular point . There is a basic mathematical explanation of
2430-418: 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
2520-414: Is based on the concept of a Hamiltonian path in graph theory . Hamilton introduced, as a method of analysis, both quaternions and biquaternions , the extension to eight dimensions by the introduction of complex number coefficients . When his work was assembled in 1853, the book Lectures on Quaternions had "formed the subject of successive courses of lectures, delivered in 1848 and subsequent years, in
2610-644: 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
2700-448: 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
2790-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing
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2880-710: The British East India Company . An expert mental calculator , the young Hamilton was capable of working out some calculations to many decimal places. In September 1813, the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, a year older than Hamilton. The two were pitted against each other in a mental arithmetic contest, with Colburn emerging as the clear victor. In reaction to his defeat, Hamilton spent less time studying languages, and more on mathematics. At age ten, he stumbled across
2970-605: The Central Bank of Ireland in 2005 to commemorate 200 years since his birth. It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized by Charles Lutwidge Dodgson in Alice in Wonderland . In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert to Euclidean geometry . In September 2022 evidence
3060-745: 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
3150-634: The Pythagorean school , whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c. AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of
3240-577: The Royal Medal of the Royal Society the following year. He was to win it again in 1848. In 1835, being secretary to the meeting of the British Association which was held that year in Dublin, Hamilton was knighted by the lord-lieutenant . Other honours rapidly succeeded, among which his election in 1837 to the president's chair in the Royal Irish Academy , and the rare distinction of being made
3330-656: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of
3420-600: 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 ), 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
3510-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of
3600-501: 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)
3690-1031: 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"):
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3780-490: 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
3870-557: The Halls of Trinity College, Dublin". Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research. When he died, Hamilton was working on a definitive statement of quaternion science. His son William Edwin Hamilton brought the Elements of Quaternions , a hefty volume of 762 pages, to publication in 1866. As copies ran short, a second edition
3960-578: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became
4050-550: 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
4140-419: 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
4230-400: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. Ancient Greek language Ancient Greek ( Ἑλληνῐκή , Hellēnikḗ ; [hellɛːnikɛ́ː] ) includes the forms of the Greek language used in ancient Greece and
4320-438: 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
4410-611: 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
4500-500: The earliest known mathematicians was Thales of Miletus ( c. 624 – c. 546 BC ); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c. 582 – c. 507 BC ) established
4590-718: The first part was printed in 1828 in the Transactions of the Royal Irish Academy . The more important contents of the second and third parts appeared in the three voluminous supplements (to the first part) which were published in the same Transactions, and in the two papers On a General Method in Dynamics , which appeared in the Philosophical Transactions in 1834 and 1835. In these papers, Hamilton developed his central principle of "Varying Action". A result of this work
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#17329098568044680-494: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of
4770-992: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,
4860-576: The founders of modern linear algebra . He has made major contributions in optics, classical mechanics , and abstract algebra . His work is fundamental to modern theoretical physics , particularly his reformulation of Newtonian mechanics . Hamiltonian mechanics including its Hamilitonian function are now central both to electromagnetism and quantum mechanics . Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819), who lived in Dublin at 29 Dominick Street , later renumbered to 36. Hamilton's father, who
4950-506: The heavens quite regularly; He left routine observation to his assistant Charles Thompson. Hamilton's sisters also supported the observatory's work. The introductory lectures by Hamilton in astronomy were celebrated; in addition to his students, they attracted scholars, poets, and women. Felicia Hemans wrote her poem The Prayer of the Lonely Student after hearing one of his lectures. Hamilton invited his four sisters to come and live at
5040-556: 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
5130-472: 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
5220-629: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"
5310-569: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at
5400-470: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to
5490-501: The last, and continued the task of finishing the Elements of Quaternions which had occupied the last six years of his life. He died on 2 September 1865, following a severe attack of gout . He is buried in Mount Jerome Cemetery in Dublin. Hamilton made outstanding contributions to classical mechanics and optics . His first discovery was in an early paper that he communicated in 1823 to John Brinkley, who presented it under
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#17329098568045580-651: The most natural medium for their description and development, and the function that is now universally known as the Hamiltonian, is the starting point for calculation in almost any area of physics. Many scientists, including Liouville , Jacobi , Darboux , Poincaré , Kolmogorov , Prigogine and Arnold , have extended Hamilton's work, in mechanics , differential equations and symplectic geometry . Hamilton's mathematical studies seem to have been undertaken and carried to their full development without collaboration, and his writings do not belong to any particular school. He
5670-533: The observatory in 1827, and they ran the household until his marriage in 1833. They included Eliza Mary Hamilton (1807–1851), the poet. In 1827, Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin. Newly appointed to the Observatory, Hamilton set off on a tour in Ireland and England with Alexander Nimmo , who was coaching him on latitude and longitude . One call
5760-499: 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
5850-454: The papers be developed further. Hamilton complied, and early in 1823, Brinkley approved the amended version. In July of 1823, Hamilton earned a place at Trinity College Dublin by examination, at age 17. His tutor there was Charles Boyton , a family friend, who brought to his attention the contemporary mathematics published by the group at the École Polytechnique in Paris. John Brinkley remarked of
5940-487: 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
6030-401: The phenomenon, namely that the wave surface is not the boundary of a convex body. A fuller understanding awaited the microlocal analysis of the middle of the 20th century, The step from optics to dynamics in the application of the method of "Varying Action" was made in 1827, and communicated to the Royal Society, in whose Philosophical Transactions for 1834 and 1835 there are two papers on
6120-434: The precocious Hamilton, "This young man, I do not say will be , but is , the first mathematician of his age." The college awarded Hamilton two optime s, or off-the-chart grades, in Greek and in physics. He was first in every subject and at every examination. He aimed to win a Trinity College fellowship by competitive examination, but this did not happen. Instead, after Brinkley in 1826 was made Bishop of Cloyne , Hamilton
6210-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in
6300-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate
6390-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced
6480-403: The side of the nearby Broom Bridge (which Hamilton called Brougham Bridge). The quaternions involved abandoning the commutative law , a radical step for the time. In the context of this prototype geometric algebra , Hamilton also introduced the cross and dot products of vector algebra, the quaternion product being the cross product minus the dot product as scalar . Hamilton also described
6570-484: The subject. Hamiltonian mechanics was a powerful new technique for working with equations of motion . Hamilton's advances enlarged the class of mechanical problems that could be solved. His principle of "Varying Action" was based on the calculus of variations , in the general class of problems included under the principle of least action which had been studied earlier by Pierre Louis Maupertuis , Euler , Joseph Louis Lagrange and others. Hamilton's analysis uncovered
6660-517: 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
6750-461: The techniques find use in electromagnetism , quantum mechanics , relativity theory and quantum field theory . In the Dictionary of Irish Biography David Spearman writes: The formulation that he devised for classical mechanics proved to be equally suited to quantum theory, whose development it facilitated. The Hamiltonian formalism shows no signs of obsolescence; new ideas continue to find this
6840-409: The title of Caustics in 1824 to the Royal Irish Academy . It was referred as usual to a committee, which recommended further development and simplification before publication. Between 1825 and 1828 the paper was expanded, and became a clearer exposition of a novel method. Over this period, Hamilton gained an appreciation for the nature and importance of optics. In 1827, Hamilton presented a theory of
6930-572: The young Hamilton to depression and illness, even to the verge of suicide. He proposed again in 1831 to Ellen de Vere, a sister of the poet Aubrey De Vere , who declined as well. Hamilton eventually married Helen Marie Bayly in 1833, a country preacher's daughter, and had three children with her: William Edwin Hamilton (born 1834), Archibald Henry (born 1835), and Helen Elizabeth (born 1840). Hamilton's married life turned out to be difficult and unhappy as Bayly proved to be pious, shy, timid, and chronically ill. Hamilton retained his faculties unimpaired to
7020-938: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in
7110-475: 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 ,
7200-508: Was appointed in 1827 to the vacant posts left by Brinkley's departure, Andrews Professor of Astronomy and Royal Astronomer of Ireland. Despite having his undergraduate career cut short in this way, he earned degrees in both classics and mathematics (BA in 1827, MA in 1837). In 1824, Hamilton was introduced at Edgeworthstown to the novelist Maria Edgeworth , by the Rev. Richard Butler , the vicar of Trim, County Meath to whom his uncle James Hamilton
7290-556: Was away from Dunsink, staying with sisters, for much of the time from 1840 to 1842. Hamilton's married life was reportedly difficult. In the troubled period of the early 1840s, his sister Sydney ran his household; when Helen returned, he was happier after some depression. Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems . Mathematicians are concerned with numbers , data , quantity , structure , space , models , and change . One of
7380-765: Was curate. During the same period, his uncle introduced him to the Disney family at Summerhill House , County Meath. The Disney sons attended Trinity College, and Hamilton had friends among them. At Summerhill, he met Catherine Disney, their sister. Hamilton was attracted to Catherine Disney, but her family did not approve and Catherine was required to marry the Rev. William Barlow, a brother of her elder sister's husband. The wedding took place in 1825. Hamilton wrote in 1826 about his feelings for her in an extended poem, "The Enthusiast". Over twenty years later, in 1847, he confided in John Herschel that during this period he might have become
7470-556: Was from Dublin, worked as a solicitor. By the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim , County Meath. Hamilton is said to have shown talent at an early age. His uncle observed that Hamilton, from a young age, had displayed an uncanny ability to acquire languages — a claim which has been disputed by some historians, who claim he had only
7560-516: Was intended by the university authorities who elected him to the Professorship of Astronomy to spend his time as he best could for the advancement of science, without restrictions. Hamilton made his discovery of the algebra of quaternions in 1843. Among much prior related work, in 1840 Benjamin Olinde Rodrigues had reached a result that amounted to their discovery in all but name. Hamilton
7650-515: Was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional Argand diagram ) to higher spatial dimensions. In working with four dimensions, rather than three, he created quaternion algebra. According to Hamilton, on 16 October he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation occurred to him; Hamilton then carved this equation using his penknife into
7740-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support
7830-459: Was prepared by Charles Jasper Joly , when the book was split into two volumes, the first appearing in 1899 and the second in 1901. The subject index and footnotes in this second edition improved the Elements accessibility. Hamilton was twice awarded the Cunningham Medal of the Royal Irish Academy . The first award, in 1834, was for his work on conical refraction, for which he also received
7920-568: Was presented to counter this suggestion, which appears to have been based on an incorrect understanding of both quaternions and their history. Hamilton married Helen Bayly, daughter of Rev Henry Bayly, Rector of Nenagh, County Tipperary, in 1833; she was a sister of neighbours to the observatory. They had three children: the journalist William Edwin Hamilton (born 1834), Archibald Henry (born 1835) and Helen Eliza Amelia (born 1840). Helen stayed with her widowed mother at Bayly Farm, Nenagh for extended periods, until her mother's death in 1837. She also
8010-579: Was the 200th anniversary of Hamilton's birth and the Irish government designated that the Hamilton Year, celebrating Irish science . Trinity College Dublin marked the year by launching the Hamilton Mathematics Institute . Two commemorative stamps were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions. A 10- euro commemorative silver proof coin was issued by
8100-459: Was to Sarah Lawrence's school at Gateacre , near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes. They visited William Wordsworth at Rydal Mount in September of that year, where the writer Caesar Otway was also present. After the visit, Hamilton sent numerous poems to Wordsworth, becoming a "poetic disciple". When Wordsworth visited Dublin in the summer of 1829, in
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