87-517: The Sand Reckoner ( Greek : Ψαμμίτης , Psammites ) is a work by Archimedes , an Ancient Greek mathematician of the 3rd century BC , in which he set out to determine an upper bound for the number of grains of sand that fit into the universe . In order to do this, Archimedes had to estimate the size of the universe according to the contemporary model, and invent a way to talk about extremely large numbers. The work, also known in Latin as Arenarius ,
174-484: A + b {\displaystyle 10^{a}10^{b}=10^{a+b}} , necessary to manipulate powers of 10. Archimedes then estimated an upper bound for the number of grains of sand required to fill the Universe. To do this, he used the heliocentric model of Aristarchus of Samos . The original work by Aristarchus has been lost. This work by Archimedes however is one of the few surviving references to his theory, whereby
261-577: A Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the world's oldest recorded living language . Among the Indo-European languages, its date of earliest written attestation is matched only by the now-extinct Anatolian languages . The Greek language is conventionally divided into the following periods: In the modern era, the Greek language entered
348-397: A positional numeral system with base 10, which is remarkable because the ancient Greeks used a very simple system for writing numbers , which employs 27 different letters of the alphabet for the units 1 through 9, the tens 10 through 90 and the hundreds 100 through 900. Archimedes also discovered and proved the law of exponents , 10 a 10 b = 10
435-594: A 4-second value for the star Vega which was a gross overestimate. The first successful stellar parallax measurements were done by Thomas Henderson in Cape Town South Africa in 1832–1833, where he measured parallax of one of the closest stars, Alpha Centauri . Between 1835 and 1836, astronomer Friedrich Georg Wilhelm von Struve at the Dorpat university observatory measured the distance of Vega , publishing his results in 1837. Friedrich Bessel ,
522-577: A basis for coinages: anthropology , photography , telephony , isomer , biomechanics , cinematography , etc. Together with Latin words , they form the foundation of international scientific and technical vocabulary ; for example, all words ending in -logy ('discourse'). There are many English words of Greek origin . Greek is an independent branch of the Indo-European language family. The ancient language most closely related to it may be ancient Macedonian , which, by most accounts,
609-478: A billion (the number of grains of sand in a dactyl-sphere) by a trillion (number of dactyl-spheres in a stadium-sphere) yields 10, the number of grains of sand in a stadium-sphere. Archimedes had estimated that the Aristarchian Universe was 10 stadia in diameter, so there would accordingly be (10) stadium-spheres in the universe, or 10. Multiplying 10 by 10 yields 10, the number of grains of sand in
696-542: A distance of tens of thousands of light-years from Earth. Data Release 2 in 2018 claims mean errors for the parallaxes of 15th magnitude and brighter stars of 20–40 microarcseconds. Very long baseline interferometry in the radio band can produce images with angular resolutions of about 1 milliarcsecond, and hence, for bright radio sources, the precision of parallax measurements made in the radio can easily exceed those of optical telescopes like Gaia. These measurements tend to be sensitivity limited, and need to be made one at
783-508: A faster, more convenient cursive writing style with the use of ink and quill . The Greek alphabet consists of 24 letters, each with an uppercase ( majuscule ) and lowercase ( minuscule ) form. The letter sigma has an additional lowercase form (ς) used in the final position of a word: In addition to the letters, the Greek alphabet features a number of diacritical signs : three different accent marks ( acute , grave , and circumflex ), originally denoting different shapes of pitch accent on
870-444: A friend of Struve, carried out an intense observational campaign in 1837–1838 at Koenigsberg Observatory for the star 61 Cygni using a heliometer , and published his results in 1838. Henderson published his results in 1839, after returning from South Africa. Those three results, two of which were measured with the best instruments at the time (Fraunhofer great refractor used by Struve and Fraunhofer heliometer by Bessel) were
957-405: A height equal to that of the highest of the mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken. But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of
SECTION 10
#17327763337021044-590: A lesser extent, in the Western Mediterranean in and around colonies such as Massalia , Monoikos , and Mainake . It was also used as the official language of government and religion in the Christian Nubian kingdoms , for most of their history. Greek, in its modern form, is the official language of Greece, where it is spoken by almost the entire population. It is also the official language of Cyprus (nominally alongside Turkish ) and
1131-400: A limited but productive system of compounding and a rich inflectional system. Although its morphological categories have been fairly stable over time, morphological changes are present throughout, particularly in the nominal and verbal systems. The major change in the nominal morphology since the classical stage was the disuse of the dative case (its functions being largely taken over by
1218-459: A mixed syllable structure, permitting complex syllabic onsets but very restricted codas. It has only oral vowels and a fairly stable set of consonantal contrasts . The main phonological changes occurred during the Hellenistic and Roman period (see Koine Greek phonology for details): In all its stages, the morphology of Greek shows an extensive set of productive derivational affixes ,
1305-404: A myriad myriads (10). Archimedes called the numbers up to 10 "first order" and called 10 itself the "unit of the second order". Multiples of this unit then became the second order, up to this unit taken a myriad-myriad times, 10·10=10. This became the "unit of the third order", whose multiples were the third order, and so on. Archimedes continued naming numbers in this way up to a myriad-myriad times
1392-605: A small number of stars. This gives more accuracy to the cosmic distance ladder and improves the knowledge of distances in the Universe, based on the dimensions of the Earth's orbit. As distances between the two points of observation are increased, the visual effect of the parallax is likewise rendered more visible. NASA 's New Horizons spacecraft performed the first interstellar parallax measurement on 22 April 2020, taking images of Proxima Centauri and Wolf 359 in conjunction with earth-based observatories. The relative proximity of
1479-460: A state of diglossia : the coexistence of vernacular and archaizing written forms of the language. What came to be known as the Greek language question was a polarization between two competing varieties of Modern Greek: Dimotiki , the vernacular form of Modern Greek proper, and Katharevousa , meaning 'purified', a compromise between Dimotiki and Ancient Greek developed in the early 19th century that
1566-408: A time, so the work is generally done only for sources like pulsars and X-ray binaries, where the radio emission is strong relative to the optical emission. Throughout the year the position of a star S is noted in relation to other stars in its apparent neighborhood: Stars that did not seem to move in relation to each other are used as reference points to determine the path of S. The observed path
1653-415: A way analogous to the way in which the orders of the first period were constructed. Continuing in this manner, he eventually arrived at the orders of the myriad-myriadth period. The largest number named by Archimedes was the last number in this period, which is Another way of describing this number is a one followed by ( short scale ) eighty quadrillion (80·10) zeroes. Archimedes' system is reminiscent of
1740-463: Is a transversal in the same (approximately Euclidean) plane as parallel lines E-v and E ′ -v, it follows that the corresponding angles of intersection of these parallel lines with this transversal are congruent: the angle θ between lines of sight E-v and E ′ -v ′ is equal to the angle θ between E ′ -v and E ′ -v ′ , which is the angle θ between observed positions of S in relation to its apparently unmoving stellar surroundings. The distance d from
1827-495: Is about eight pages long in translation and is addressed to the Syracusan king Gelo II (son of Hiero II ). It is considered the most accessible work of Archimedes. First, Archimedes had to invent a system of naming large numbers . The number system in use at that time could express numbers up to a myriad (μυριάς — 10,000), and by utilizing the word myriad itself, one can immediately extend this to naming all numbers up to
SECTION 20
#17327763337021914-515: Is an ellipse: the projection of Earth's orbit around the Sun through S onto the distant background of non-moving stars. The farther S is removed from Earth's orbital axis, the greater the eccentricity of the path of S. The center of the ellipse corresponds to the point where S would be seen from the Sun: The plane of Earth's orbit is at an angle to a line from the Sun through S. The vertices v and v' of
2001-417: Is approximately that subtended by an object 2 centimeters in diameter located 5.3 kilometers away. For a right triangle , where p {\displaystyle p} is the parallax, 1 au (149,600,000 km) is approximately the average distance from the Sun to Earth, and d {\displaystyle d} is the distance to the star. Using small-angle approximations (valid when
2088-415: Is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its magnitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the Earth, including in it all the seas and the hollows of the Earth filled up to
2175-826: Is sometimes called aljamiado , as when Romance languages are written in the Arabic alphabet. Article 1 of the Universal Declaration of Human Rights in Greek: Transcription of the example text into Latin alphabet : Article 1 of the Universal Declaration of Human Rights in English: Proto-Greek Mycenaean Ancient Koine Medieval Modern Stellar parallax Stellar parallax
2262-763: Is still used internationally for the writing of Ancient Greek . In Greek, the question mark is written as the English semicolon, while the functions of the colon and semicolon are performed by a raised point (•), known as the ano teleia ( άνω τελεία ). In Greek the comma also functions as a silent letter in a handful of Greek words, principally distinguishing ό,τι ( ó,ti , 'whatever') from ότι ( óti , 'that'). Ancient Greek texts often used scriptio continua ('continuous writing'), which means that ancient authors and scribes would write word after word with no spaces or punctuation between words to differentiate or mark boundaries. Boustrophedon , or bi-directional text,
2349-429: Is the apparent shift of position ( parallax ) of any nearby star (or other object) against the background of distant stars. By extension, it is a method for determining the distance to the star through trigonometry, the stellar parallax method . Created by the different orbital positions of Earth , the extremely small observed shift is largest at time intervals of about six months, when Earth arrives at opposite sides of
2436-632: Is to modern spoken English ". Greek is spoken today by at least 13 million people, principally in Greece and Cyprus along with a sizable Greek-speaking minority in Albania near the Greek-Albanian border. A significant percentage of Albania's population has knowledge of the Greek language due in part to the Albanian wave of immigration to Greece in the 1980s and '90s and the Greek community in
2523-562: The British Overseas Territory of Akrotiri and Dhekelia (alongside English ). Because of the membership of Greece and Cyprus in the European Union, Greek is one of the organization's 24 official languages . Greek is recognized as a minority language in Albania, and used co-officially in some of its municipalities, in the districts of Gjirokastër and Sarandë . It is also an official minority language in
2610-657: The European canon . Greek is also the language in which many of the foundational texts in science and philosophy were originally composed. The New Testament of the Christian Bible was also originally written in Greek. Together with the Latin texts and traditions of the Roman world , the Greek texts and Greek societies of antiquity constitute the objects of study of the discipline of Classics . During antiquity , Greek
2697-516: The Indo-Iranian languages (see Graeco-Aryan ), but little definitive evidence has been found. In addition, Albanian has also been considered somewhat related to Greek and Armenian, and it has been proposed that they all form a higher-order subgroup along with other extinct languages of the ancient Balkans; this higher-order subgroup is usually termed Palaeo-Balkan , and Greek has a central position in it. Linear B , attested as early as
The Sand Reckoner - Misplaced Pages Continue
2784-514: The Sun remains unmoved while the Earth orbits the Sun. In Archimedes's own words: His [Aristarchus'] hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes
2871-780: The United Kingdom , and throughout the European Union , especially in Germany . Historically, significant Greek-speaking communities and regions were found throughout the Eastern Mediterranean , in what are today Southern Italy , Turkey , Cyprus , Syria , Lebanon , Israel , Palestine , Egypt , and Libya ; in the area of the Black Sea , in what are today Turkey, Bulgaria , Romania , Ukraine , Russia , Georgia , Armenia , and Azerbaijan ; and, to
2958-443: The aberration of light and the nutation of Earth's axis, and catalogued 3,222 stars. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars. In the second quarter of the 19th century, technological progress reached to the level which provided sufficient accuracy and precision for stellar parallax measurements. Giuseppe Calandrelli noted stellar parallax in 1805-6 and came up with
3045-468: The genitive ). The verbal system has lost the infinitive , the synthetically -formed future, and perfect tenses and the optative mood . Many have been replaced by periphrastic ( analytical ) forms. Pronouns show distinctions in person (1st, 2nd, and 3rd), number (singular, dual , and plural in the ancient language; singular and plural alone in later stages), and gender (masculine, feminine, and neuter), and decline for case (from six cases in
3132-417: The syntax of Greek have remained constant: verbs agree with their subject only, the use of the surviving cases is largely intact (nominative for subjects and predicates, accusative for objects of most verbs and many prepositions, genitive for possessors), articles precede nouns, adpositions are largely prepositional, relative clauses follow the noun they modify and relative pronouns are clause-initial. However,
3219-566: The 20th century on), especially from French and English, are typically not inflected; other modern borrowings are derived from Albanian , South Slavic ( Macedonian / Bulgarian ) and Eastern Romance languages ( Aromanian and Megleno-Romanian ). Greek words have been widely borrowed into other languages, including English. Example words include: mathematics , physics , astronomy , democracy , philosophy , athletics , theatre, rhetoric , baptism , evangelist , etc. Moreover, Greek words and word elements continue to be productive as
3306-498: The Aristarchian Universe. Following Archimedes's estimate of a myriad (10,000) grains of sand in a poppy seed; 64,000 poppy seeds in a dactyl-sphere; the length of a stadium as 10,000 dactyls; and accepting 19mm as the width of a dactyl, the diameter of Archimedes's typical sand grain would be 18.3 μm, which today we would call a grain of silt . Currently, the smallest grain of sand would be defined as 50 μm in diameter. Archimedes made some interesting experiments and computations along
3393-532: The Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface. The reason for the large size of this model is that the Greeks were unable to observe stellar parallax with available techniques, which implies that any parallax is extremely small and so the stars must be placed at great distances from the Earth (assuming heliocentrism to be true). According to Archimedes, Aristarchus did not state how far
3480-544: The Greek alphabet since approximately the 9th century BC. It was created by modifying the Phoenician alphabet , with the innovation of adopting certain letters to represent the vowels. The variant of the alphabet in use today is essentially the late Ionic variant, introduced for writing classical Attic in 403 BC. In classical Greek, as in classical Latin, only upper-case letters existed. The lower-case Greek letters were developed much later by medieval scribes to permit
3567-549: The Greek language was the Cypriot syllabary (also a descendant of Linear A via the intermediate Cypro-Minoan syllabary ), which is closely related to Linear B but uses somewhat different syllabic conventions to represent phoneme sequences. The Cypriot syllabary is attested in Cyprus from the 11th century BC until its gradual abandonment in the late Classical period, in favor of the standard Greek alphabet. Greek has been written in
The Sand Reckoner - Misplaced Pages Continue
3654-593: The Indo-European language family. It is native to Greece , Cyprus , Italy (in Calabria and Salento ), southern Albania , and other regions of the Balkans , Caucasus , the Black Sea coast, Asia Minor , and the Eastern Mediterranean . It has the longest documented history of any Indo-European language, spanning at least 3,400 years of written records. Its writing system is the Greek alphabet , which has been used for approximately 2,800 years; previously, Greek
3741-535: The Sun Diameter of Earth {\displaystyle {\frac {\text{Diameter of Universe}}{\text{Diameter of Earth orbit around the Sun}}}={\frac {\text{Diameter of Earth orbit around the Sun}}{\text{ Diameter of Earth}}}} In order to obtain an upper bound, Archimedes made the following assumptions of their dimensions: Archimedes then concluded that the diameter of the Universe
3828-543: The Sun in its orbit, giving a baseline distance of about two astronomical units between observations. The parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit (AU). Stellar parallax is so difficult to detect that its existence was the subject of much debate in astronomy for hundreds of years. Thomas Henderson , Friedrich Georg Wilhelm von Struve , and Friedrich Bessel made
3915-415: The Sun to S now follows from simple trigonometry: so that d = E-Sun / tan( 1 / 2 θ), where E-Sun is 1 AU. The more distant an object is, the smaller its parallax. Stellar parallax measures are given in the tiny units of arcseconds , or even in thousandths of arcseconds (milliarcseconds). The distance unit parsec is defined as the length of the leg of a right triangle adjacent to
4002-448: The Sun), whose parallax is 0.7685, is 1 / 0.7685 parsecs = 1.301 parsecs (4.24 ly) distant. Stellar parallax is most often measured using annual parallax , defined as the difference in position of a star as seen from Earth and Sun, i.e. the angle subtended at a star by the mean radius of Earth's orbit around the Sun. The parsec (3.26 light-years ) is defined as the distance for which
4089-404: The acute during the late 20th century, and it has only been retained in typography . After the writing reform of 1982, most diacritics are no longer used. Since then, Greek has been written mostly in the simplified monotonic orthography (or monotonic system), which employs only the acute accent and the diaeresis. The traditional system, now called the polytonic orthography (or polytonic system),
4176-481: The angle is small compared to 1 radian ), so the parallax, measured in arcseconds, is If the parallax is 1", then the distance is This defines the parsec , a convenient unit for measuring distance using parallax. Therefore, the distance, measured in parsecs, is simply d = 1 / p {\displaystyle d=1/p} , when the parallax is given in arcseconds. Precise parallax measurements of distance have an associated error. This error in
4263-584: The angle of one arcsecond at one vertex , where the other leg is 1 AU long. Because stellar parallaxes and distances all involve such skinny right triangles , a convenient trigonometric approximation can be used to convert parallaxes (in arcseconds) to distance (in parsecs). The approximate distance is simply the reciprocal of the parallax: d (pc) ≈ 1 / p (arcsec) . {\displaystyle d{\text{ (pc)}}\approx 1/p{\text{ (arcsec)}}.} For example, Proxima Centauri (the nearest star to Earth other than
4350-434: The annual parallax is 1 arcsecond . Annual parallax is normally measured by observing the position of a star at different times of the year as Earth moves through its orbit. The angles involved in these calculations are very small and thus difficult to measure. The nearest star to the Sun (and also the star with the largest parallax), Proxima Centauri , has a parallax of 0.7685 ± 0.0002 arcsec. This angle
4437-479: The arrival of Proto-Greeks, some documented in Mycenaean texts ; they include a large number of Greek toponyms . The form and meaning of many words have changed. Loanwords (words of foreign origin) have entered the language, mainly from Latin, Venetian , and Turkish . During the older periods of Greek, loanwords into Greek acquired Greek inflections, thus leaving only a foreign root word. Modern borrowings (from
SECTION 50
#17327763337024524-796: The country. Prior to the Greco-Turkish War and the resulting population exchange in 1923 a very large population of Greek-speakers also existed in Turkey , though very few remain today. A small Greek-speaking community is also found in Bulgaria near the Greek-Bulgarian border. Greek is also spoken worldwide by the sizable Greek diaspora which has notable communities in the United States , Australia , Canada , South Africa , Chile , Brazil , Argentina , Russia , Ukraine ,
4611-412: The cube of a linear dimension ("For it has been proved that spheres have the triplicate ratio to one another of their diameters") then a sphere one dactyl in diameter would contain (using our current number system) 40, or 64,000 poppy seeds. He then claimed (without evidence) that each poppy seed could contain a myriad (10,000) grains of sand. Multiplying the two figures together he proposed 640,000,000 as
4698-441: The dative led to a rise of prepositional indirect objects (and the use of the genitive to directly mark these as well). Ancient Greek tended to be verb-final, but neutral word order in the modern language is VSO or SVO. Modern Greek inherits most of its vocabulary from Ancient Greek, which in turn is an Indo-European language, but also includes a number of borrowings from the languages of the populations that inhabited Greece before
4785-415: The different distances between the viewer and the Sun, whether viewed from the center of the Earth or from the surface of the Earth at sunrise. This may be the first known computation dealing with solar parallax. There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which
4872-445: The distance of the movement of the Earth compared to the distance to these infinitely far away stars is, within the accuracy of the measurement, 0. Thus a line of sight from Earth's first position E to vertex v will be essentially the same as a line of sight from the Earth's second position E ′ to the same vertex v, and will therefore run parallel to it - impossible to depict convincingly in an image of limited size: Since line E ′ -v ′
4959-528: The earliest forms attested to four in the modern language). Nouns, articles, and adjectives show all the distinctions except for a person. Both attributive and predicative adjectives agree with the noun. The inflectional categories of the Greek verb have likewise remained largely the same over the course of the language's history but with significant changes in the number of distinctions within each category and their morphological expression. Greek verbs have synthetic inflectional forms for: Many aspects of
5046-416: The elliptical projection of the path of S are projections of positions of Earth E and E ′ such that a line E-E ′ intersects the line Sun-S at a right angle; the triangle created by points E, E ′ and S is an isosceles triangle with the line Sun-S as its symmetry axis. Any stars that did not move between observations are, for the purpose of the accuracy of the measurement, infinitely far away. This means that
5133-479: The end of the 19th century, mostly by use of the filar micrometer . Astrographs using astronomical photographic plates sped the process in the early 20th century. Automated plate-measuring machines and more sophisticated computer technology of the 1960s allowed more efficient compilation of star catalogues . In the 1980s, charge-coupled devices (CCDs) replaced photographic plates and reduced optical uncertainties to one milliarcsecond. Stellar parallax remains
5220-412: The extent that one can speak of a new language emerging. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language. It is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, " Homeric Greek is probably closer to Demotic than 12-century Middle English
5307-450: The first ones in history to establish the reliable distance scale to the stars. A large heliometer was installed at Kuffner Observatory (In Vienna) in 1896, and was used for measuring the distance to other stars by trigonometric parallax. By 1910 it had computed 16 parallax distances to other stars, out of only 108 total known to science at that time. Being very difficult to measure, only about 60 stellar parallaxes had been obtained by
SECTION 60
#17327763337025394-513: The first successful parallax measurements in 1832–1838, for the stars Alpha Centauri , Vega , and 61 Cygni . Stellar parallax is so small that it was unobservable until the 19th century, and its apparent absence was used as a scientific argument against heliocentrism during the early modern age . It is clear from Euclid 's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons, such gigantic distances involved seemed entirely implausible: it
5481-449: The language show both conservative and innovative tendencies across the entire attestation of the language from the ancient to the modern period. The division into conventional periods is, as with all such periodizations, relatively arbitrary, especially because, in all periods, Ancient Greek has enjoyed high prestige, and the literate borrowed heavily from it. Across its history, the syllabic structure of Greek has varied little: Greek shows
5568-467: The late 15th century BC, was the first script used to write Greek. It is basically a syllabary , which was finally deciphered by Michael Ventris and John Chadwick in the 1950s (its precursor, Linear A , has not been deciphered and most likely encodes a non-Greek language). The language of the Linear B texts, Mycenaean Greek , is the earliest known form of Greek. Another similar system used to write
5655-452: The many other countries of the Greek diaspora . Greek roots have been widely used for centuries and continue to be widely used to coin new words in other languages; Greek and Latin are the predominant sources of international scientific vocabulary . Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, or possibly earlier. The earliest written evidence is
5742-422: The mass of sand equal in magnitude to the Earth filled up in the way described, but also that of the mass equal in magnitude to the universe. Greek language Greek ( Modern Greek : Ελληνικά , romanized : Elliniká , [eliniˈka] ; Ancient Greek : Ἑλληνική , romanized : Hellēnikḗ ) is an Indo-European language, constituting an independent Hellenic branch within
5829-399: The measured parallax angle does not translate directly into an error for the distance, except for relatively small errors. The reason for this is that an error toward a smaller angle results in a greater error in distance than an error toward a larger angle. However, an approximation of the distance error can be computed by where d is the distance and p is the parallax. The approximation
5916-465: The morphological changes also have their counterparts in the syntax, and there are also significant differences between the syntax of the ancient and that of the modern form of the language . Ancient Greek made great use of participial constructions and of constructions involving the infinitive, and the modern variety lacks the infinitive entirely (employing a raft of new periphrastic constructions instead) and uses participles more restrictively. The loss of
6003-417: The number of grains of sand that would fit in the universe. A Greek stadium had a length of 600 Greek feet, and each foot was 16 dactyls long, so there were 9,600 dactyls in a stadium. Archimedes rounded this number up to 10,000 (a myriad) to make calculations easier, again, noting that the resulting number will exceed the actual number of grains of sand. The cube of 10,000 is a trillion (10); and multiplying
6090-511: The number of hypothetical grains of sand in a sphere one dactyl in diameter. To make further calculations easier, he rounded up 640 million to one billion, noting only that the first number is smaller than the second, and that therefore the number of grains of sand calculated subsequently will exceed the actual number of grains. Recall that Archimedes's meta-goal with this essay was to show how to calculate with what were previously considered impossibly large numbers, not simply to accurately calculate
6177-492: The number of stellar parallaxes measured to milliarcsecond accuracy a thousandfold. Even so, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away, a little more than one percent of the diameter of the Milky Way Galaxy . The Hubble telescope WFC3 now has a precision of 20 to 40 microarcseconds, enabling reliable distance measurements up to 3,066 parsecs (10,000 ly) for
6264-643: The regions of Apulia and Calabria in Italy. In the framework of the European Charter for Regional or Minority Languages , Greek is protected and promoted officially as a regional and minority language in Armenia, Hungary , Romania, and Ukraine. It is recognized as a minority language and protected in Turkey by the 1923 Treaty of Lausanne . The phonology , morphology , syntax , and vocabulary of
6351-431: The standard for calibrating other measurement methods (see Cosmic distance ladder ). Accurate calculations of distance based on stellar parallax require a measurement of the distance from Earth to the Sun, now known to exquisite accuracy based on radar reflection off the surfaces of planets. In 1989, the satellite Hipparcos was launched primarily for obtaining parallaxes and proper motions of nearby stars, increasing
6438-421: The stars were from the Earth. Archimedes therefore had to make the following assumptions: This assumption can also be expressed by saying that the stellar parallax caused by the motion of the Earth around its orbit equals the solar parallax caused by motion around the Earth. Put in a ratio: Diameter of Universe Diameter of Earth orbit around the Sun = Diameter of Earth orbit around
6525-437: The stressed vowel; the so-called breathing marks ( rough and smooth breathing ), originally used to signal presence or absence of word-initial /h/; and the diaeresis , used to mark the full syllabic value of a vowel that would otherwise be read as part of a diphthong. These marks were introduced during the course of the Hellenistic period. Actual usage of the grave in handwriting saw a rapid decline in favor of uniform usage of
6612-463: The two stars combined with the 6.5 billion kilometer (about 43 AU) distance of the spacecraft from Earth yielded a discernible parallax of arcminutes, allowing the parallax to be seen visually without instrumentation. The European Space Agency 's Gaia mission , launched 19 December 2013, is expected to measure parallax angles to an accuracy of 10 micro arcseconds for all moderately bright stars, thus mapping nearby stars (and potentially planets) up to
6699-421: The unit of the 10-th order, i.e., (10)^(10) After having done this, Archimedes called the orders he had defined the "orders of the first period", and called the last one, ( 10 8 ) ( 10 8 ) {\displaystyle (10^{8})^{(10^{8})}} , the "unit of the second period". He then constructed the orders of the second period by taking multiples of this unit in
6786-494: The way. One experiment was to estimate the angular size of the Sun, as seen from the Earth. Archimedes's method is especially interesting as it takes into account the finite size of the eye's pupil, and therefore may be the first known example of experimentation in psychophysics , the branch of psychology dealing with the mechanics of human perception , whose development is generally attributed to Hermann von Helmholtz . Another interesting computation accounts for solar parallax and
6873-574: Was a distinct dialect of Greek itself. Aside from the Macedonian question, current consensus regards Phrygian as the closest relative of Greek, since they share a number of phonological, morphological and lexical isoglosses , with some being exclusive between them. Scholars have proposed a Graeco-Phrygian subgroup out of which Greek and Phrygian originated. Among living languages, some Indo-Europeanists suggest that Greek may be most closely related to Armenian (see Graeco-Armenian ) or
6960-505: Was also used in Ancient Greek. Greek has occasionally been written in the Latin script , especially in areas under Venetian rule or by Greek Catholics . The term Frankolevantinika / Φραγκολεβαντίνικα applies when the Latin script is used to write Greek in the cultural ambit of Catholicism (because Frankos / Φράγκος is an older Greek term for West-European dating to when most of (Roman Catholic Christian) West Europe
7047-556: Was by far the most widely spoken lingua franca in the Mediterranean world . It eventually became the official language of the Byzantine Empire and developed into Medieval Greek . In its modern form , Greek is the official language of Greece and Cyprus and one of the 24 official languages of the European Union . It is spoken by at least 13.5 million people today in Greece, Cyprus, Italy, Albania, Turkey , and
7134-518: Was no more than 10 stadia (in modern units, about 2 light years ), and that it would require no more than 10 grains of sand to fill it. With these measurements, each grain of sand in Archimedes's thought-experiment would have been approximately 19 μm (0.019 mm) in diameter. Archimedes claims that forty poppy-seeds laid side by side would equal one Greek dactyl (finger-width) which was approximately 19 mm (3/4 inch) in length. Since volume proceeds as
7221-437: Was one of Tycho Brahe 's principal objections to Copernican heliocentrism that for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere (the fixed stars). James Bradley first tried to measure stellar parallaxes in 1729. The stellar movement proved too insignificant for his telescope , but he instead discovered
7308-485: Was recorded in writing systems such as Linear B and the Cypriot syllabary . The alphabet arose from the Phoenician script and was in turn the basis of the Latin , Cyrillic , Coptic , Gothic , and many other writing systems. The Greek language holds a very important place in the history of the Western world. Beginning with the epics of Homer , ancient Greek literature includes many works of lasting importance in
7395-580: Was under the control of the Frankish Empire ). Frankochiotika / Φραγκοχιώτικα (meaning 'Catholic Chiot') alludes to the significant presence of Catholic missionaries based on the island of Chios . Additionally, the term Greeklish is often used when the Greek language is written in a Latin script in online communications. The Latin script is nowadays used by the Greek-speaking communities of Southern Italy . The Yevanic dialect
7482-637: Was used for literary and official purposes in the newly formed Greek state. In 1976, Dimotiki was declared the official language of Greece, after having incorporated features of Katharevousa and thus giving birth to Standard Modern Greek , used today for all official purposes and in education . The historical unity and continuing identity between the various stages of the Greek language are often emphasized. Although Greek has undergone morphological and phonological changes comparable to those seen in other languages, never since classical antiquity has its cultural, literary, and orthographic tradition been interrupted to
7569-673: Was written by Romaniote and Constantinopolitan Karaite Jews using the Hebrew Alphabet . In a tradition, that in modern time, has come to be known as Greek Aljamiado , some Greek Muslims from Crete wrote their Cretan Greek in the Arabic alphabet . The same happened among Epirote Muslims in Ioannina . This also happened among Arabic-speaking Byzantine rite Christians in the Levant ( Lebanon , Palestine, and Syria ). This usage
#701298