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157-626: Based on the palaeography and orthography of the hieratic text, the text was most likely written down in the 13th Dynasty and based on older material probably dating to the Twelfth Dynasty of Egypt , roughly 1850 BC. Approximately 5.5 m (18 ft) long and varying between 3.8 and 7.6 cm (1.5 and 3 in) wide, its format was divided by the Soviet Orientalist Vasily Vasilievich Struve in 1930 into 25 problems with solutions. It

314-409: A Persian astronomer and mathematician , correctly computed the fractional part of 2 π to 9 sexagesimal digits in 1424, and translated this into 16 decimal digits after the decimal point: which gives 16 correct digits for π after the decimal point: He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 2 sides. In the second half of the 16th century,

471-438: A compromise between the beauty and legibility of the capitals and the rapidity of the cursive, and is clearly an artificial product. It was certainly in existence by the latter part of the 4th century, for a number of manuscripts of that date are written in perfect uncial hands ( Exempla , pl. XX). It presently supplanted the capitals and appears in numerous manuscripts which have survived from the 5th, 6th and 7th centuries, when it

628-579: A few more centuries. In north India, Prakrit was replaced by Sanskrit by the end of the 3rd century, while this change took place about a century later in south India. Some of the inscriptions though written in Prakrit, were influenced by Sanskrit and vice versa. The epigraphs of the Kushana kings are found in a mixture of Prakrit and Sanskrit, while the Mathura inscriptions of the time of Sodasa, belonging to

785-402: A general resemblance (with considerable differences in detail) both to the minuscule cursive of late papyri, and to those used in modern Greek type; uncial forms were avoided. In the course of the 10th century the hand, without losing its beauty and exactness, gained in freedom. Its finest period was from the 9th to the 12th century, after which it rapidly declined. The development was marked by

942-419: A hand more precisely than a range of at least seventy or eighty years". In a 2005 e-mail addendum to his 1996 "The Paleographical Dating of P-46" paper Bruce W. Griffin stated "Until more rigorous methodologies are developed, it is difficult to construct a 95% confidence interval for [ New Testament ] manuscripts without allowing a century for an assigned date." William Schniedewind went even further in

1099-499: A hierarchy of texts in a suite through discourse analysis, determining the provenance of texts, identifying forgeries , interpolations and recensions with precision; eliciting a professional authenticity in documentation, textual and manuscript evaluation with view to producing a critical edition if required and a critical assessment of a given discourse event as rendered and set in a materiality or medium. Knowledge of writing materials and discourse material production systems

1256-459: A lesser extent ⟨A⟩ ( ⟨α⟩ ). The earliest Greek papyrus yet discovered is probably that containing the Persae of Timotheus , which dates from the second half of the 4th century BC and its script has a curiously archaic appearance. ⟨E⟩ , ⟨Σ⟩ , and ⟨Ω⟩ have the capital form, and apart from these test letters the general effect

1413-415: A minuscule hand. Although the characteristic forms of the uncial type appear to have their origin in the early cursive, the two hands are nevertheless quite distinct. The uncial is a libraria , closely related to the capital writing, from which it differs only in the rounding off of the angles of certain letters, principally [REDACTED] [REDACTED] [REDACTED] [REDACTED] . It represents

1570-596: A mistake in the 528th decimal place, and that all succeeding digits were incorrect. In the early years of the computer, an expansion of π to 100 000 decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the aforementioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating

1727-402: A new type of hand, the minuscule , which originated in the 8th century, as an adaptation to literary purposes of the second of the types of Byzantine cursive mentioned above. A first attempt at a calligraphic use of this hand, seen in one or two manuscripts of the 8th or early 9th century, in which it slopes to the right and has a narrow, angular appearance, did not find favour, but by the end of

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1884-520: A number of inscriptions belonging to the dynasties of Pallava, Chola and Pandya are found. These records are written in three different scripts known as Tamil , Vattezhuttu and Grantha scripts , the last variety being used to write Sanskrit inscriptions. In the Kerala region, the Vattezhuttu script developed into a still more cursive script called Kolezhuthu during the 14th and 15th centuries. At

2041-461: A pen. In cutting a hard surface, it is easier to form angles than curves; in writing the reverse is the case; hence the development of writing was from angular letters ("capitals") inherited from epigraphic style to rounded ones ("uncials"). But only certain letters were affected by this development, in particular ⟨E⟩ (uncial ⟨ε⟩ ), ⟨Σ⟩ ( ⟨c⟩ ), ⟨Ω⟩ ( ⟨ω⟩ ), and to

2198-617: A precision of less than one Planck length (at 1.6162 × 10   meters , the shortest unit of length expected to be directly measurable) using π expressed to just 62 decimal places. The English amateur mathematician William Shanks calculated π to 530 decimal places in January 1853, of which the first 527 were correct (the last few likely being incorrect due to round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced right at

2355-432: A rather broken appearance, part of one character being often made separately from the rest and linked to the next letter. A form characteristic of the 1st and 2nd century and surviving after that only as a fraction sign (= 1 ⁄ 8 ) is ⟨η⟩ in the shape [REDACTED] . By the end of the 1st century, there had been developed several excellent types of cursive, which, though differing considerably both in

2512-502: A register in each given dialect and language has evolved constantly, it is necessary to know how to decipher its individual substantive, occurrence make-up and constituency. For example, assessing its characters and typology as they existed in various places, times and locations. In addition, for hand-written texts, scribes often use many abbreviations , and annotations so as to functionally aid speed, efficiency and ease of writing and in some registers to importantly save invaluable space of

2669-561: A similar series found at Verespatak in Transylvania ( CIL , iii) and a number of papyri. From a study of a number of documents which exhibit transitional forms, it appears that this cursive was originally simplified capital writing. The evolution was so rapid, however, that at quite an early date the scriptura epistolaris of the Roman world can no longer be described as capitals. By the 1st century, this kind of writing began to develop

2826-404: A single curve. The cursive writing thus foreshadows the specifically uncial forms. The same specimens show great inequality in the height of the letters; the main strokes are prolonged upwards ( [REDACTED] = ⟨b⟩ ; [REDACTED] = ⟨d⟩ ) or downwards ( [REDACTED] = ⟨q⟩ ; [REDACTED] = 's ). In this direction, the cursive tends to become

2983-456: A tendency But from the first there were several styles, varying from the formal, regular hands characteristic of service books to the informal style, marked by numerous abbreviations, used in manuscripts intended only for a scholar's private use. The more formal hands were exceedingly conservative, and there are few classes of script more difficult to date than the Greek minuscule of this class. In

3140-410: A text, document or manuscript; and analysis of the substantive textual content of documents is a secondary function. Included in the discipline is the practice of deciphering, reading, and dating manuscripts, and the cultural context of writing, including the methods with which writing and printing of texts, manuscripts, books , codices and tomes, tracts and monographs , etcetera, were produced, and

3297-453: Is 20,000. Approximating π to four decimal places: π ≈ 62832 ⁄ 20000 = 3.1416, Aryabhata stated that his result "approximately" ( āsanna "approaching") gave the circumference of a circle. His 15th-century commentator Nilakantha Somayaji ( Kerala school of astronomy and mathematics ) has argued that the word means not only that this is an approximation, but that the value is incommensurable (irrational) . Further progress

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3454-434: Is a gradual decline in the subcontinent of such disciplines as palaeography, epigraphy and numismatics . The discipline of ancient Indian scripts and the languages they are written needs new scholars who, by adopting traditional palaeographic methods and modern technology, may decipher, study and transcribe the various types of epigraphs and legends still extant today. The language of the earliest written records, that is,

3611-407: Is a round, upright hand seen, for example, in a British Museum papyrus containing Odyssey III. The cross-stroke of ⟨ε⟩ is high, ⟨Μ⟩ deeply curved and ⟨Α⟩ has the form ⟨α⟩ . Uniformity of size is well attained, and a few strokes project, and these but slightly, above or below the line. Another type, well called by palaeographer Schubart

3768-632: Is a well-known mathematical papyrus, usually referenced together with the Rhind Mathematical Papyrus . The Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two. The problems in the Moscow Papyrus follow no particular order, and the solutions of the problems provide much less detail than those in the Rhind Mathematical Papyrus . The papyrus

3925-514: Is accurate to two sexagesimal digits. The Chinese mathematician Liu Hui in 263 CE computed π to between 3.141 024 and 3.142 708 by inscribing a 96-gon and 192-gon; the average of these two values is 3.141 866 (accuracy 9·10 ). He also suggested that 3.14 was a good enough approximation for practical purposes. He has also frequently been credited with a later and more accurate result, π ≈ 3927 ⁄ 1250 = 3.1416 (accuracy 2·10 ), although some scholars instead believe that this

4082-497: Is called the Byzantine period, that is, roughly from AD 300 to 650, is known as the biblical hand. It went back to at least the end of the 2nd century and had had originally no special connection with Christian literature . In both vellum and paper manuscripts from 4th-century Egypt are other forms of script, particularly a sloping, rather inelegant hand derived from the literary hand of the 3rd century, which persisted until at least

4239-459: Is carried to an extreme. In a letter of the prefect, dated in 209, we have a fine example of the Chancery hand , with tall and laterally compressed letters, ⟨ο⟩ very narrow and ⟨α⟩ and ⟨ω⟩ often written high in the line. This style, from at least the latter part of the 2nd century, exercised considerable influence on the local hands, many of which show

4396-411: Is carried very far, the linking of letters reaching the point of illegibility, and the characters sloping to the right. ⟨A⟩ is reduced to a mere acute angle ( ⟨∠⟩ ), ⟨T⟩ has the cross-stroke only on the left, ⟨ω⟩ becomes an almost straight line, ⟨H⟩ acquires a shape somewhat like h , and the last stroke of ⟨N⟩

4553-546: Is difficult to trace, as there are few examples, mostly not datable on external grounds. Only for the 3rd century BC have we a secure basis. The hands of that period have an angular appearance; there is little uniformity in the size of individual letters, and though sometimes, notably in the Petrie papyrus containing the Phaedo of Plato , a style of considerable delicacy is attained, the book-hand in general shows less mastery than

4710-455: Is due to the later (5th-century) Chinese mathematician Zu Chongzhi . Zu Chongzhi is known to have computed π to be between 3.1415926 and 3.1415927, which was correct to seven decimal places. He also gave two other approximations of π : π ≈ 22 ⁄ 7 and π ≈ 355 ⁄ 113 , which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation of π using fewer than five decimal digits in

4867-401: Is extended far upwards and at times flattened out until it is little more than a diagonal stroke to the right. The attempt to secure a horizontal line along the top is here abandoned. This style was not due to inexpertness, but to the desire for speed, being used especially in accounts and drafts, and was generally the work of practised writers. How well established the cursive hand had now become

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5024-435: Is foundational to the study of handwriting and printing events and to the identification of the periods in which a document or manuscript may have been produced. An important goal may be to assign the text a date and a place of origin, or determining which translations of a text are produced from which specific document or manuscript. This is why the palaeographer and attendant semiologists and philologists must take into account

5181-527: Is known as the Cave character , and their script differs from the Northern version in being more angular. Most of the modern scripts of South India have evolved from this script, with the exception of Vatteluttu , the exact origins of which are unknown, and Nandinagari , which is a variant of Devanagari that developed due to later Northern influence. In south India from the 7th century of the common era onwards,

5338-494: Is made by the few other papyri, chiefly literary, dating from about 300 BC; ⟨E⟩ may be slightly rounded, ⟨Ω⟩ approach the uncial form, and the angular ⟨Σ⟩ occurs as a letter only in the Timotheus papyrus, though it survived longer as a numeral (= 200), but the hands hardly suggest that for at least a century and a half the art of writing on papyrus had been well established. Yet before

5495-480: Is not much later, the letters are larger and more heavily made; in the 5th-century Alexandrinus , a later development is seen with emphatic distinction of thick and thin strokes. By the 6th century, alike in vellum and in papyrus manuscripts, the heaviness had become very marked, though the hand still retained, in its best examples, a handsome appearance; but after this it steadily deteriorated, becoming ever more mechanical and artificial. The thick strokes grew heavier;

5652-521: Is of great interest as being the ancestor of the type called (from its later occurrence in vellum codices of the Bible ) the biblical hand. This, which can be traced back at least the late 2nd century, has a square, rather heavy appearance; the letters, of uniform size, stand upright, and thick and thin strokes are well distinguished. In the 3rd century the book-hand, like the cursive, appears to have deteriorated in regularity and stylistic accomplishment. In

5809-405: Is one of stiffness and angularity. More striking is the hand of the earliest dated papyrus, a contract of 311 BC. Written with more ease and elegance, it shows little trace of any development towards a truly cursive style; the letters are not linked, and though the uncial ⟨c⟩ is used throughout, ⟨E⟩ and ⟨Ω⟩ have the capital forms. A similar impression

5966-495: Is opposed to the minuscule, based on a system of four lines, with letters of unequal height, adpl . Another classification, according to the care taken in forming the letters, distinguishes between the set book-hand and the cursive script. The difference in this case is determined by the subject matter of the text; the writing used for books ( scriptura libraria ) is in all periods quite distinct from that used for letters and documents ( epistolaris , diplomatica ). While

6123-493: Is shown in some wax tablets of this period, the writing on which, despite the difference of material, closely resemble the hands of papyri. Documents of the late 3rd and early 2nd centuries BC show there is nothing analogous to the Apollonius letters, perhaps partly by the accident of survival. In the more formal types the letters stand rather stiffly upright, often without the linking strokes, and are more uniform in size; in

6280-1068: Is the matres lectionis system to indicate certain vowels. Early Phoenician-derived scripts did not have letters for vowels, and so most texts recorded just consonants. Most likely as a consequence of phonetic changes in North Semitic languages, the Aramaeans reused certain letters in the alphabet to represent long vowels. The letter aleph was employed to write /ā/, he for /ō/, yod for /ī/, and vav for /ū/. Aramaic writing and language supplanted Babylonian cuneiform and Akkadian language , even in their homeland in Mesopotamia . The wide diffusion of Aramaic letters led to its writing being used not only in monumental inscriptions, but also on papyrus and potsherds . Aramaic papyri have been found in large numbers in Egypt, especially at Elephantine —among them are official and private documents of

6437-409: Is the study and academic discipline of the analysis of historical writing systems, the historicity of manuscripts and texts, subsuming deciphering and dating of historical manuscripts, including the analysis of historic penmanship , handwriting script , signification and printed media . It is primarily concerned with the forms, processes and relationships of writing and printing systems as evident in

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6594-511: Is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively. The remaining problems are more common in nature. Problems 2 and 3 are ship's part problems. One of the problems calculates the length of a ship's rudder and the other computes the length of a ship's mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long. Aha problems involve finding unknown quantities (referred to as aha , "stack") if

6751-559: The Ancient Middle East , originating in what is modern-day Syria , between 1000 and 600 BC. It spread from the Mediterranean coast to the borders of India, becoming extremely popular and being adopted by many people, both with or without any previous writing system. The Aramaic script was written in a consonantal form with a direction from right to left. The Aramaic alphabet , a modified form of Phoenician ,

6908-480: The Augustan Age . Epigraphists divide the numerous inscriptions of this period into two quite distinct classes: tituli , or formal inscriptions engraved on stone in elegant and regular capitals, and acta , or legal texts, documents, etc., generally engraved on bronze in cramped and careless capitals. Palaeography inherits both these types. Reproduced by scribes on papyrus or parchment, the elegant characters of

7065-653: The Edicts of Ashoka , is Prakrit . Besides Prakrit, the Ashokan edicts are also written in Greek and Aramaic. Moreover, all the edicts of Ashoka engraved in the Kharoshthi and Brahmi scripts are in the Prakrit language: thus, originally the language employed in the inscriptions was Prakrit, with Sanskrit adopted at a later stage. Past the period of the Maurya Empire , the use of Prakrit continued in inscriptions for

7222-736: The Greek alphabet were also added to the Indian context after its penetration in the early centuries AD, with the Arabic alphabet following in the 13th century. After a lapse of a few centuries the Kharoṣṭhi script became obsolete; the Greek script in India went through a similar fate and disappeared. But the Brahmi and Arabic scripts endured for a much longer period. Moreover, there was a change and development in

7379-552: The Jewish military settlement in 5 BC. In the Aramaic papyri and potsherds, words are separated usually by a small gap, as in modern writing. At the turn of the 3rd to 2nd centuries BC, the heretofore uniform Aramaic letters developed new forms, as a result of dialectal and political fragmentation in several subgroups. The most important of these is the so-called square Hebrew block script , followed by Palmyrene , Nabataean , and

7536-646: The Second Intermediate Period , c. 1600 BCE, although stated to be a copy of an older, Middle Kingdom text) implies an approximation of π as 256 ⁄ 81 ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle via approximation with the octagon . Astronomical calculations in the Shatapatha Brahmana (c. 6th century BCE) use a fractional approximation of 339 ⁄ 108 ≈ 3.139 . The Mahabharata (500 BCE – 300 CE) offers an approximation of 3, in

7693-557: The University of Tokyo computed π to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of π . In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000 , a 64-node supercomputer with 1 terabyte of main memory, to calculate π to roughly 1.24 trillion digits in around 600 hours (25   days). Depending on

7850-637: The round basin located in front of the Temple in Jerusalem as having a diameter of 10 cubits and a circumference of 30 cubits. The issue is discussed in the Talmud and in Rabbinic literature . Among the many explanations and comments are these: There is still some debate on this passage in biblical scholarship. Many reconstructions of the basin show a wider brim (or flared lip) extending outward from

8007-549: The unit fraction having that number as denominator , e.g. 4 ¯ = 1 4 {\displaystyle {\bar {4}}={\frac {1}{4}}} ; unit fractions were common objects of study in ancient Egyptian mathematics. Other mathematical texts from Ancient Egypt include: General papyri: For the 2/n tables see: Palaeography Palaeography ( UK ) or paleography ( US ; ultimately from ‹See Tfd› Greek : παλαιός , palaiós , 'old', and γράφειν , gráphein , 'to write')

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8164-528: The volume of a truncated pyramid : where a and b are the base and top side lengths of the truncated pyramid and h is the height. Researchers have speculated how the Egyptians might have arrived at the formula for the volume of a frustum but the derivation of this formula is not given in the papyrus. Richard J. Gillings gave a cursory summary of the Papyrus' contents. Numbers with overlines denote

8321-488: The " Indiana Pi Bill " of 1897, which stated "the ratio of the diameter and circumference is as five-fourths to four" (which would imply " π = 3.2 ") and a passage in the Hebrew Bible that implies that π = 3 . The so-called "Indiana Pi Bill" from 1897 has often been characterized as an attempt to "legislate the value of Pi". Rather, the bill dealt with a purported solution to the problem of geometrically " squaring

8478-411: The "severe" style, has a more angular appearance and not infrequently slopes to the right; though handsome, it has not the sumptuous appearance of the former. There are various classes of a less pretentious style, in which convenience rather than beauty was the first consideration and no pains were taken to avoid irregularities in the shape and alignment of the letters. Lastly may be mentioned a hand which

8635-640: The 10th century onwards. The use of Nandinagari , a variant of Nagari script, is mostly confined to the Karnataka region. In central India, mostly in Madhya Pradesh , the inscriptions of the Vakatakas , and the kings of Sarabhapura and Kosala were written in what are known as "box-headed" and "nail-headed" characters. It may be noted that the early Kadambas of Karnataka also employed "nail-headed" characters in some of their inscriptions. During

8792-566: The 10th to 8th centuries BC, especially extensive state treaties ( c.  750 BC ) and royal inscriptions. The early Old Ancient should be classified as "Ancient Aramaic" and consists of two clearly distinguished and standardised written languages, the Early Ancient Aramaic and the Late Ancient Aramaic. Aramaic was influenced at first principally by Akkadian , then from the 5th century BC by Persian and from

8949-453: The 10th, 11th and 12th centuries a sloping hand, less dignified than the upright, formal type, but often very handsome, was especially used for manuscripts of the classics. Hands of the 11th century are marked in general (though there are exceptions) by a certain grace and delicacy, exact but easy; those of the 12th by a broad, bold sweep and an increasing freedom, which readily admits uncial forms, ligatures and enlarged letters but has not lost

9106-411: The 19th century such scholars as Wilhelm Wattenbach , Leopold Delisle and Ludwig Traube contributed greatly to making palaeography independent from diplomatic. In the 20th century, the "New French School" of palaeographers, especially Jean Mallon , gave a new direction to the study of scripts by stressing the importance of ductus (the shape and order of the strokes used to compose letters) in studying

9263-460: The 1st century, are to be found in manuscripts of Virgil and Terence . Neither of these forms of capital writing offers any difficulty in reading, except that no space is left between the words. Their dates are still uncertain, in spite of attempts to determine them by minute observation. The rustic capitals, more practical than the square forms, soon came into general use. This was the standard form of writing, so far as books are concerned, until

9420-546: The 3rd century BC onwards by Greek , as well as by Hebrew , especially in Palestine . As Aramaic evolved into the imperial language of the Neo-Assyrian Empire , the script used to write it underwent a change into something more cursive. The best examples of this script come from documents written on papyrus from Egypt. About 500 BC, Darius I (522–486) made the Aramaic used by the imperial administration into

9577-520: The 3rd century BCE, Archimedes proved the sharp inequalities 223 ⁄ 71  <  π  <  22 ⁄ 7 , by means of regular 96-gons (accuracies of 2·10 and 4·10 , respectively). In the 2nd century CE, Ptolemy used the value 377 ⁄ 120 , the first known approximation accurate to three decimal places (accuracy 2·10 ). It is equal to 3 + 8 / 60 + 30 / 60 2 , {\displaystyle 3+8/60+30/60^{2},} which

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9734-534: The 3rd–4th century, the script used in the inscriptions of Ikshvakus of Nagarjunakonda developed a unique style of letter-forms with elongated verticals and artistic flourishes, which did not continue after their rule. The earliest attested form of writing in South India is represented by inscriptions found in caves, associated with the Chalukya and Chera dynasties. These are written in variants of what

9891-487: The 4th century AD, it was replaced by Greek, Persian, the eastern and western dialects of Aramaic and Arabic, though not without leaving its traces in the written form of most of these. In its original Achaemenid form, Imperial Aramaic is found in texts of the 5th to 3rd centuries BC. These come mostly from Egypt and especially from the Jewish military colony of Elephantine , which existed at least from 530 to 399 BC. A history of Greek handwriting must be incomplete owing to

10048-626: The 4th century onwards, with the rise of the Guptas, Sanskrit became the predominant language of India and continued to be employed in texts and inscriptions of all parts of India along with the regional languages in the subsequent centuries. The copper-plate charters of the Pallavas , the Cholas and the Pandyas documents are written in both Sanskrit and Tamil. Kannada is used in texts dating from about

10205-424: The 530th decimal place rendered the rest of his calculation erroneous; due to the nature of Machin's formula, the error propagated back to the 528th decimal place, leaving only the first 527 digits correct once again. Twenty years later, Shanks expanded his calculation to 707 decimal places in April 1873. Due to this being an expansion of his previous calculation, most of the new digits were incorrect as well. Shanks

10362-595: The 5th century and the Halmidi inscription is considered to be the earliest epigraph written in the Kannada language . Inscriptions in Telugu began to appear from the 6th or 7th century. Malayalam made its beginning in writings from the 15th century onwards. In north India, the Brahmi script was used over a vast area; however, Ashokan inscriptions are also found using Kharoshthi , Aramaic and Greek scripts . With

10519-454: The 5th century, from which unfortunately few dated documents have survived. Byzantine cursive tends to an exuberant hand, in which the long strokes are excessively extended and individual letters often much enlarged. But not a few hands of the 5th and 6th centuries are truly handsome and show considerable technical accomplishment. Both an upright and a sloping type occur and there are many less ornamental hands, but there gradually emerged towards

10676-447: The 5th century, when it was replaced by a new type, the uncial, which is discussed below. While the set book-hand, in square or rustic capitals, was used for the copying of books, the writing of everyday life, letters and documents of all kinds, was in a cursive form, the oldest examples of which are provided by the graffiti on walls at Pompeii ( CIL , iv), a series of waxen tablets, also discovered at Pompeii ( CIL , iv, supplement),

10833-406: The 5th century. Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century ( Ludolph van Ceulen ), and 126 digits by the 19th century ( Jurij Vega ), surpassing

10990-578: The 5th century. The three great early codices of the Bible are all written in uncials of the biblical type. In the Vaticanus , placed during the 4th century, the characteristics of the hand are least strongly marked; the letters have the forms characteristic of the type but without the heavy appearance of later manuscripts, and the general impression is one of greater roundness. In the Sinaiticus , which

11147-523: The 6th to about the 10th century, the inscriptions in north India were written in a script variously named, e.g., Siddhamatrika and Kutila ("Rañjanā script"). From the 8th century, Siddhamatrika developed into the Śāradā script in Kashmir and Punjab , into Proto-Bengali or Gaudi in Bengal and Orissa , and into Nagari in other parts of north India. Nagari script was used widely in northern India from

11304-429: The 7th century two general types, one (especially used in letters and contracts) a current hand, sloping to the right, with long strokes in such characters at ⟨τ⟩ , ⟨ρ⟩ , ⟨ξ⟩ , ⟨η⟩ (which has the h shape), ⟨ι⟩ , and ⟨κ⟩ , and with much linking of letters, and another (frequent in accounts), which shows, at least in essence, most of

11461-472: The 8th century, though with some differences from modern practice. At no period down to the invention of printing did Greek scribes consistently separate words. The book-hand of papyri aimed at an unbroken succession of letters, except for distinction of sections; in cursive hands, especially where abbreviations were numerous, some tendency to separate words may be recognised, but in reality it was phrases or groups of letters rather than words which were divided. In

11618-496: The 9th century a more ornamental type, from which modern Greek script descended, was already established. It has been suggested that it was evolved in the Monastery of Stoudios at Constantinople . In its earliest examples it is upright and exact but lacks flexibility; accents are small, breathings square in formation, and in general only such ligatures are used as involve no change in the shape of letters. The single forms have

11775-512: The Brahmi script which may be traced in time and space through the Maurya , Kuṣaṇa , Gupta and early medieval periods. The present-day Nāgarī script is derived from Brahmi. The Brahmi is also the ancestral script of most other Indian scripts, in northern and southern South Asia. Legends and inscriptions in Brahmi are engraved upon leather, wood, terracotta, ivory, stone, copper, bronze, silver and gold. Arabic got an important place, particularly in

11932-614: The Festal letters despatched annually by the Patriarch of Alexandria , was occasionally used, the best known example being the Codex Marchalianus (6th or 7th century). A combination of this hand with the other type is also known. The uncial hand lingered on, mainly for liturgical manuscripts, where a large and easily legible script was serviceable, as late as the 12th century, but in ordinary use it had long been superseded by

12089-459: The French mathematician François Viète discovered an infinite product that converged on π known as Viète's formula . The German-Dutch mathematician Ludolph van Ceulen ( circa 1600) computed the first 35 decimal places of π with a 2 -gon. He was so proud of this accomplishment that he had them inscribed on his tombstone . In Cyclometricus (1621), Willebrord Snellius demonstrated that

12246-471: The Moscow Mathematical calculates the volume of a frustum . Problem 14 states that a pyramid has been truncated in such a way that the top area is a square of length 2 units, the bottom a square of length 4 units, and the height 6 units, as shown. The volume is found to be 56 cubic units, which is correct. The text of the example runs like this: "If you are told: a truncated pyramid of 6 for

12403-585: The Palestinian sheikh, Toubias, are in a type of script which cannot be very unlike the Chancery hand of the time, and show the Ptolemaic cursive at its best. These hands have a noble spaciousness and strength, and though the individual letters are by no means uniform in size there is a real unity of style, the general impression being one of breadth and uprightness. ⟨H⟩ , with the cross-stroke high, ⟨Π⟩ , ⟨Μ⟩ , with

12560-530: The Slovene mathematician Jurij Vega improved John Machin 's formula to calculate the first 140 digits, of which the first 126 were correct. In 1841, William Rutherford calculated 208 digits, of which the first 152 were correct. The magnitude of such precision (152 decimal places) can be put into context by the fact that the circumference of the largest known object, the observable universe, can be calculated from its diameter (93   billion light-years ) to

12717-454: The West, where Greek scribes were in request to produce manuscripts of the classical authors, there was a revival, and several manuscripts of this period, though markedly inferior to those of the 11th and 12th centuries, are by no means without beauty. In the book-hand of early papyri, neither accents nor breathings were employed. Their use was established by the beginning of the Roman period, but

12874-425: The abstract to his 2005 paper "Problems of Paleographic Dating of Inscriptions" and stated: "The so-called science of paleography often relies on circular reasoning because there is insufficient data to draw precise conclusion about dating. Scholars also tend to oversimplify diachronic development, assuming models of simplicity rather than complexity". The Aramaic language was the international trade language of

13031-405: The accuracy required for any conceivable application outside of pure mathematics. The record of manual approximation of π is held by William Shanks , who calculated 527 decimals correctly in 1853. Since the middle of the 20th century, the approximation of π has been the task of electronic digital computers (for a comprehensive account, see Chronology of computation of π ). On June 28, 2024,

13188-639: The advent of the Saka - Kshatrapas and the Kushanas as political powers in north India, the writing system underwent a definite change due to the use of new writing tools and techniques. Further development of the Brahmi script and perceivable changes in its evolutionary trend can be discerned during the Gupta period: in fact, the Gupta script is considered to be the successor of the Kushana script in north India. From

13345-400: The area of a semi-cylinder (Peet). Below we assume that the problem refers to the area of a hemisphere. The text of problem 10 runs like this: "Example of calculating a basket. You are given a basket with a mouth of 4 1/2. What is its surface? Take 1/9 of 9 (since) the basket is half an egg-shell. You get 1. Calculate the remainder which is 8. Calculate 1/9 of 8. You get 2/3 + 1/6 + 1/18. Find

13502-440: The beginning of the line. The coronis , a more elaborate form of this, marked the beginning of lyrics or the principal sections of a longer work. Punctuation marks, the comma , the high, low and middle points, were established in the book-hand by the Roman period; in early Ptolemaic papyri, a double point ( ⟨:⟩ ) is found. In vellum and paper manuscripts, punctuation marks and accents were regularly used from at least

13659-492: The book-hand was occasionally used for documents. Since the scribe did not date literary rolls, such papyri are useful in tracing the development of the book-hand. The documents of the mid-3rd century BC show a great variety of cursive hands. There are none from chancelleries of the Hellenistic monarchs, but some letters, notably those of Apollonius , the finance minister of Ptolemy II , to this agent, Zeno, and those of

13816-467: The bowl itself by several inches to match the description given in NRSV In the succeeding verses, the rim is described as "a handbreadth thick; and the brim thereof was wrought like the brim of a cup, like the flower of a lily: it received and held three thousand baths" NRSV , which suggests a shape that can be encompassed with a string shorter than the total length of the brim, e.g., a Lilium flower or

13973-497: The charred rolls found at Herculaneum are specimens of Greek literary hands from outside Egypt dating to c.  1 AD . A comparison with the Egyptian papyri reveals great similarity in style and shows that conclusions drawn from the henads of Egypt may, with caution, be applied to the development of writing in the Greek world generally. The cursive hand of the 4th century shows some uncertainty of character. Side by side with

14130-517: The circle ". The bill was nearly passed by the Indiana General Assembly in the U.S., and has been claimed to imply a number of different values for π , although the closest it comes to explicitly asserting one is the wording "the ratio of the diameter and circumference is as five-fourths to four", which would make π = 16 ⁄ 5 = 3.2 , a discrepancy of nearly 2 percent. A mathematics professor who happened to be present

14287-406: The contemporary cursive. In the 2nd century, the letters grew rounder and more uniform in size, but in the 1st century there is a certain disintegration perceptible, as in the cursive hand. Probably at no time did the Ptolemaic book-hand acquire such unity of stylistic effect as the cursive. Papyri of the Roman period are far more numerous and show greater variety. The cursive of the 1st century has

14444-435: The cross strokes of ⟨T⟩ and ⟨Θ⟩ and the base of ⟨Δ⟩ were furnished with drooping spurs. The hand, which is often singularly ugly , passed through various modifications, now sloping, now upright, though it is not certain that these variations were really successive rather than concurrent. A different type of uncials, derived from the Chancery hand and seen in two papyrus examples of

14601-592: The current record was established by the StorageReview Lab team with Alexander Yee's y-cruncher with 202 trillion (2.02× 10 ) digits. The best known approximations to π dating to before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics in particular by the mid-first millennium, to an accuracy of seven decimal places. After this, no further progress

14758-519: The day the bill was brought up for consideration in the Senate, after it had passed in the House, helped to stop the passage of the bill on its second reading, after which the assembly thoroughly ridiculed it before postponing it indefinitely . It is sometimes claimed that the Hebrew Bible implies that " π equals three", based on a passage in 1 Kings 7:23 and 2 Chronicles 4:2 giving measurements for

14915-521: The description of Solomon's Temple in the Hebrew Bible ). The Babylonians were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of π as 25 ⁄ 8 = 3.125, about 0.528% below the exact value. At about the same time, the Egyptian Rhind Mathematical Papyrus (dated to

15072-459: The digits of π . For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of π were published in 1962. The authors outlined what would be needed to calculate π to 1 million decimal places and concluded that

15229-449: The establishment of the great Alexandrian Library , which systematically copied literary and scientific works, and to the multifarious activities of Hellenistic bureaucracy. From here onward, the two types of script were sufficiently distinct (though each influenced the other) to require separate treatment. Some literary papyri, like the roll containing Aristotle 's Constitution of Athens , were written in cursive hands, and, conversely,

15386-460: The first dominated by majuscule and the second by minuscule writing. Jean Mabillon , a French Benedictine monk, scholar and antiquary , whose work De re diplomatica was published in 1681, is widely regarded as the founder of the twin disciplines of palaeography and diplomatics . However, the actual term "palaeography" was coined (in Latin) by Bernard de Montfaucon , a Benedictine monk , in

15543-657: The first quarter of the 1st century, contain verses in classical Sanskrit. From the 4th century onwards, the Gupta Empire came to power and supported the Sanskrit language and literature. In western India and also in some regions of Andhra Pradesh and Karnataka , Prakrit was used till the 4th century, mostly in the Buddhist writings though in a few contemporary records of the Ikshvakus of Nagarjunakonda , Sanskrit

15700-400: The form of two almost perpendicular strokes joined only at the top, are usually small; ⟨ω⟩ is rather flat, its second loop reduced to a practically straight line. Partly by the broad flat tops of the larger letters, partly by the insertion of a stroke connecting those (like H, Υ) which are not naturally adapted to linking, the scribes produced the effect of a horizontal line along

15857-458: The forms of individual letters and in general appearance, bear a family likeness to one another. Qualities which are specially noticeable are roundness in the shape of letters, continuity of formation, the pen being carried on from character to character, and regularity, the letters not differing strikingly in size and projecting strokes above or below the line being avoided. Sometimes, especially in tax-receipts and in stereotyped formulae, cursiveness

16014-439: The forms of letters as followed that from metal to papyrus. The justification for considering the two materials separately is that after the general adoption of vellum, the Egyptian evidence is first supplemented and later superseded by that of manuscripts from elsewhere, and that during this period the hand most used was one not previously employed for literary purposes. The prevailing type of book-hand during what in papyrology

16171-459: The forms of the later minuscule. (cf. below .) This is often upright, though a slope to the right is quite common, and sometimes, especially in one or two documents of the early Arabic period, it has an almost calligraphic effect. In the Byzantine period, the book-hand, which in earlier times had more than once approximated to the contemporary cursive, diverged widely from it. The change from papyrus to vellum involved no such modification in

16328-578: The formularies of the private documents and the Proverbs of Ahiqar have maintained an older tradition of sentence structure and style. Imperial Aramaic immediately replaced Ancient Aramaic as a written language and, with slight modifications, it remained the official, commercial and literary language of the Near East until gradually, beginning with the fall of the Achaemenids in 331 BC and ending in

16485-473: The fragmentary nature of evidence. If one rules out the inscriptions on stone or metal, which belong to the science of epigraphy , there is practically a dependence on papyri from Egypt for the period preceding the 4th or 5th century AD, the earliest of which take back our knowledge only to the end of the 4th century BC. This limitation is less serious than might appear, since the few manuscripts not of Egyptian origin which have survived from this period, like

16642-491: The historical development of scripts. The Latin alphabet first appears in the epigraphic type of majuscule writing, known as capitals. These characters form the main stem from which developed all the branches of Latin writing. On the oldest monuments (the inscriptiones bello Hannibalico antiquiores of the Corpus Inscriptionum Latinarum = CIL ), it is far from showing the orderly regularity of

16799-438: The historical usages of various styles of handwriting, common writing customs, and scribal or notarial abbreviations, annotations conventions, annexures , addenda and specifics of printed typology, syntagm and proxemics must be assessed as a collective undertaking. Philological knowledge of the register, language, vocabulary, and grammar generally used at a given time, place and circumstance may assist palaeographers to identify

16956-428: The history of scriptoria . This discipline is important for understanding, authenticating, and dating historic texts. However, in the absence of additional evidence, it cannot be used to pinpoint exact dates. The discipline is one of the auxiliary sciences of history , and is considered to have been founded by Jean Mabillon with his work De re diplomatica , published in 1681, which was the first textbook to address

17113-426: The identity 1 4 π = 4 arccot ⁡ 5 − arccot ⁡ 239 {\textstyle {\tfrac {1}{4}}\pi =4\operatorname {arccot} 5-\operatorname {arccot} 239} to calculate 100 digits of π (see § Machin-like formula below). In 1719, Thomas de Lagny used a similar identity to calculate 127 digits (of which 112 were correct). In 1789,

17270-419: The inscriptions become the square capitals of the manuscripts, and the actuaria , as the writing of the acta is called, becomes the rustic capital . Of the many books written in square capitals, the éditions de luxe of ancient times, only a few fragments have survived, the most famous being pages from manuscripts of Virgil . The finest examples of rustic capitals, the use of which is attested by papyri of

17427-468: The later hands, which become progressively stiff and affected. Approximations of %CF%80 Approximations for the mathematical constant pi ( π ) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era . In Chinese mathematics , this was improved to approximations correct to what corresponds to about seven decimal digits by

17584-421: The later minuscule word-division is much commoner but never became systematic, accents and breathings serving of themselves to indicate the proper division. The view that the art of writing in India developed gradually, as in other areas of the world, by going through the stages of pictographic, ideographic and transitional phases of the phonetic script, which in turn developed into syllabic and alphabetic scripts

17741-428: The later period. Side by side with upright and square characters are angular and sloping forms, sometimes very distorted, which seem to indicate the existence of an early cursive writing from which they would have been borrowed. Certain literary texts clearly allude to such a hand. Later, the characters of the cursive type were progressively eliminated from formal inscriptions, and capital writing reached its perfection in

17898-453: The loss of the feeling for style. A fortunate accident has preserved two Greek parchments written in Parthia , one dated 88 BC, in a practically unligatured hand, the other, 22/21 BC, in a very cursive script of Ptolemaic type; and though each has non-Egyptian features the general character indicates a uniformity of style in the Hellenistic world. The development of the Ptolemaic book-hand

18055-473: The materials used. To this general rule the Greek script is no exception. Whatever may have been the period at which the use of papyrus or leather as a writing material began in Greece (and papyrus was employed in the 5th century BC), it is highly probable that for some time after the introduction of the alphabet the characters were incised with a sharp tool on stones or metal far oftener than they were written with

18212-407: The medium. Hence, the specialist-palaeographer, philologist and semiotician must know how to, in the broadest sense, interpret, comprehend and understand them. Knowledge of individual letterforms , typographic ligatures , signs, typology , fonts , graphemes, hieroglyphics , and signification forms in general, subsuming punctuation , syntagm and proxemics , abbreviations and annotations; enables

18369-426: The middle of the 3rd century BC, one finds both a practised book-hand and a developed and often remarkably handsome cursive. These facts may be due to accident, the few early papyri happening to represent an archaic style which had survived along with a more advanced one; but it is likely that there was a rapid development at this period, due partly to the opening of Egypt, with its supplies of papyri, and still more to

18526-476: The middle stroke reduced to a very shallow curve, sometimes approaching a horizontal line, ⟨Υ⟩ , and ⟨Τ⟩ , with its cross-bar extending much further to the left than to the right of the up-stroke, ⟨Γ⟩ and ⟨Ν⟩ , whose last stroke is prolonged upwards above the line, often curving backwards, are all broad; ⟨ε⟩ , ⟨c⟩ , ⟨θ⟩ and ⟨β⟩ , which sometimes takes

18683-415: The more cursive they are apt to be packed closely together. These features are more marked in the hands of the 2nd century. The less cursive often show am approximation to the book-hand, the letters growing rounder and less angular than in the 3rd century; in the more cursive linking was carried further, both by the insertion of coupling strokes and by the writing of several letters continuously without raising

18840-485: The much later Syriac script . Aramaic is usually divided into three main parts: The term Middle Aramaic refers to the form of Aramaic which appears in pointed texts and is reached in the 3rd century AD with the loss of short unstressed vowels in open syllables, and continues until the triumph of Arabic . Old Aramaic appeared in the 11th century BC as the official language of the first Aramaean states . The oldest witnesses to it are inscriptions from northern Syria of

18997-407: The numerator and denominator. Zu Chongzhi's results surpass the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium. In Gupta-era India (6th century), mathematician Aryabhata , in his astronomical treatise Āryabhaṭīya stated: Add 4 to 100, multiply by 8 and add to 62,000. This is 'approximately' the circumference of a circle whose diameter

19154-498: The official language of the western half of the Achaemenid Empire . This so-called " Imperial Aramaic " (the oldest dated example, from Egypt, belonging to 495 BC) is based on an otherwise unknown written form of Ancient Aramaic from Babylonia . In orthography, Imperial Aramaic preserves historical forms— alphabet , orthography , morphology , pronunciation , vocabulary , syntax and style are highly standardised. Only

19311-521: The output of a shoemaker given that he has to cut and decorate sandals. Seven of the twenty-five problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere (problem 10) and finding the volume of a frustum (a truncated pyramid). The tenth problem of the Moscow Mathematical Papyrus asks for a calculation of the surface area of a hemisphere (Struve, Gillings) or possibly

19468-564: The outset to certain fundamental definitions and principles of the science. The original characters of an alphabet are modified by the material and the implements used. When stone and chisel are discarded for papyrus and reed-pen, the hand encounters less resistance and moves more rapidly. This leads to changes in the size and position of the letters, and then to the joining of letters, and, consequently, to altered shapes. We are thus confronted at an early date with quite distinct types. The majuscule style of writing, based on two parallel lines, ADPL ,

19625-465: The palaeographer to read, comprehend and then to understand the text and/or the relationship and hierarchy between texts in suite. The palaeographer, philologist and semiotician must first determine language, then dialect and then the register, function and purpose of the text. That is, one must by necessity become expert in the formation, historicity and evolution of these languages and signification communities, and material communication events . Secondly,

19782-484: The parchments from Avroman or Dura , the Herculaneum papyri , and a few documents found in Egypt but written elsewhere, reveal a uniformity of style in the various portions of the Greek world; however, differences can be discerned, with it being probable that distinct local styles could be traced were there more material to analyze. Further, during any given period several types of hand may exist together. There

19939-450: The pen, so that before the end of the century an almost current hand was evolved. A characteristic letter, which survived into the early Roman period, is ⟨T⟩ , with its cross-stroke made in two portions (variants: [REDACTED] ). In the 1st century, the hand tended, so far as can be inferred from surviving examples, to disintegrate; one can recognise the signs which portend a change of style, irregularity, want of direction, and

20096-1672: The perimeter of the inscribed polygon converges on the circumference twice as fast as does the perimeter of the corresponding circumscribed polygon. This was proved by Christiaan Huygens in 1654. Snellius was able to obtain seven digits of π from a 96-sided polygon . In 1656, John Wallis published the Wallis product : π 2 = ∏ n = 1 ∞ 4 n 2 4 n 2 − 1 = ∏ n = 1 ∞ ( 2 n 2 n − 1 ⋅ 2 n 2 n + 1 ) = ( 2 1 ⋅ 2 3 ) ⋅ ( 4 3 ⋅ 4 5 ) ⋅ ( 6 5 ⋅ 6 7 ) ⋅ ( 8 7 ⋅ 8 9 ) ⋅ ⋯ {\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdot \;\cdots } In 1706, John Machin used Gregory's series (the Taylor series for arctangent ) and

20253-402: The principal characteristics of two new types: the uncial and the minuscule cursive . With the coming into use of writing surfaces which were smooth, or offered little resistance, the unhampered haste of the writer altered the shape, size and position of the letters. In the earliest specimens of writing on wax, plaster or papyrus, there appears a tendency to represent several straight strokes by

20410-518: The problems are pefsu problems (see: Egyptian algebra ): 10 of the 25 problems. A pefsu measures the strength of the beer made from a hekat of grain A higher pefsu number means weaker bread or beer. The pefsu number is mentioned in many offering lists. For example, problem 8 translates as: Problems 11 and 23 are Baku problems. These calculate the output of workers. Problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to? Problem 23 finds

20567-539: The purpose of a calculation, π can be approximated by using fractions for ease of calculation. The most notable such approximations are 22 ⁄ 7 ( relative error of about 4·10 ) and 355 ⁄ 113 (relative error of about 8·10 ). In Chinese mathematics, the fractions 22/7 and 355/113 are known as Yuelü ( 约率 ; yuēlǜ ; 'approximate ratio') and Milü ( 密率 ; mìlǜ ; 'close ratio'). Of some notability are legal or historical texts purportedly "defining π " to have some rational value, such as

20724-480: The ratios offered in Bhishma Parva verses: 6.12.40–45. ... The Moon is handed down by memory to be eleven thousand yojanas in diameter. Its peripheral circle happens to be thirty three thousand yojanas when calculated. ... The Sun is eight thousand yojanas and another two thousand  yojanas in diameter. From that its peripheral circle comes to be equal to thirty thousand yojanas. ... In

20881-486: The remainder of this 8 after subtracting 2/3 + 1/6 + 1/18. You get 7 + 1/9. Multiply 7 + 1/9 by 4 + 1/2. You get 32. Behold this is its area. You have found it correctly." The solution amounts to computing the area as The formula calculates for the area of a hemisphere, where the scribe of the Moscow Papyrus used 256 81 ≈ 3.16049 {\displaystyle {\frac {256}{81}}\approx 3.16049} to approximate π . The fourteenth problem of

21038-566: The royalty, during the medieval period and it provides rich material for history writing. The decipherment and subsequent development of Indus glyphs is also a matter for continuing research and discussion. Most of the available inscriptions and manuscripts written in the above scripts—in languages like Prakrit , Pali , Sanskrit , Apabhraṃśa , Tamil and Persian —have been read and exploited for history writing, but numerous inscriptions preserved in different museums still remain undeciphered for lack of competent palaeographic Indologists, as there

21195-407: The same characteristics less pronounced; and its effects may be traced into the early part of the 4th century. Hands of the 3rd century uninfluenced by it show a falling off from the perfection of the 2nd century; stylistic uncertainty and a growing coarseness of execution mark a period of decline and transition. Several different types of book-hand were used in the Roman period. Particularly handsome

21352-604: The same time, the modern Malayalam script developed out of the Grantha script. The early form of the Telugu-Kannada script is found in the inscriptions of the early Kadambas of Banavasi and the early Chalukyas of Badami in the west, and Salankayana and the early Eastern Chalukyas in the east who ruled the Kannada and Telugu speaking areas respectively, during the 4th to 7th centuries. Attention should be drawn at

21509-416: The sense of style and decorative effect. In the 13th and still more in the 14th centuries there was a steady decline; the less formal hands lost their beauty and exactness, becoming ever more disorderly and chaotic in their effect, while formal style imitated the precision of an earlier period without attaining its freedom and naturalness, and often appears singularly lifeless. In the 15th century, especially in

21666-464: The series. His series are now the basis for the fastest algorithms currently used to calculate π . Evaluating the first term alone yields a value correct to seven decimal places: See Ramanujan–Sato series . From the mid-20th century onwards, all improvements in calculation of π have been done with the help of calculators or computers . In 1944−45, D. F. Ferguson, with the aid of a mechanical desk calculator , found that William Shanks had made

21823-484: The set book-hand, in majuscule or minuscule, shows a tendency to stabilise the forms of the letters, the cursive, often carelessly written, is continually changing in the course of years and according to the preferences of the writers. This being granted, a summary survey of the morphological history of the Latin alphabet shows the zenith of its modifications at once, for its history is divided into two very unequal periods,

21980-401: The style founded on the Chancery hand , regular in formation and with tall and narrow letters, which characterised the period of Diocletian , and lasted well into the century, we find many other types mostly marked by a certain looseness and irregularity. A general progress towards a florid and sprawling hand is easily recognisable, but a consistent and deliberate style was hardly evolved before

22137-508: The style, substance and formation of the text, document and manuscript and the handwriting style and printed typology, grapheme typos and lexical and signification system(s) employed. Palaeography may be employed to provide information about the date at which a document was written. However, "paleography is a last resort for dating" and, "for book hands, a period of 50 years is the least acceptable spread of time" with it being suggested that "the 'rule of thumb' should probably be to avoid dating

22294-473: The subject. The term palaeography was coined by Bernard de Montfaucon with the publication of his work on Greek palaeography, the Palaeographia Graeca, in 1708. Palaeography is an essential skill for many historians , semioticians and philologists , as it addresses a suite of interrelated lines of inquiry. First, since the style of an alphabet , grapheme or sign system set within

22451-497: The sum of the quantity and part(s) of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance, problem 19 asks one to calculate a quantity taken 1 + 1 ⁄ 2 times and added to 4 to make 10. In other words, in modern mathematical notation one is asked to solve 3 2 x + 4 = 10 {\displaystyle {\frac {3}{2}}x+4=10} . Most of

22608-459: The task was beyond that day's technology, but would be possible in five to seven years. In 1989, the Chudnovsky brothers computed π to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of π : Records since then have all been accomplished using the Chudnovsky algorithm . In 1999, Yasumasa Kanada and his team at

22765-562: The title of his Palaeographia Graeca (1708), which remained a standard work in the specific field of Greek palaeography for more than a century. With their establishment of palaeography, Mabillon and his fellow Benedictines were responding to the Jesuit Daniel Papebroch , who doubted the authenticity of some of the documents which the Benedictines offered as credentials for the authorisation of their monasteries. In

22922-422: The top of the writing, from which the letters seem to hang. This feature is indeed a general characteristic of the more formal Ptolemaic script, but it is specially marked in the 3rd century BC. Besides these hand of Chancery type, there are numerous less elaborate examples of cursive, varying according to the writer's skill and degree of education, and many of them strikingly easy and handsome. In some cursiveness

23079-427: The vertical height by 4 on the base by 2 on the top: You are to square the 4; result 16. You are to double 4; result 8. You are to square this 2; result 4. You are to add the 16 and the 8 and the 4; result 28. You are to take 1/3 of 6; result 2. You are to take 28 twice; result 56. See, it is of 56. You will find [it] right" The solution to the problem indicates that the Egyptians knew the correct formula for obtaining

23236-405: Was a marked difference between the hand used for literary works (generally called " uncials " but, in the papyrus period, better styled "book-hand") and that of documents (" cursive ") and within each of these classes several distinct styles were employed side by side; and the various types are not equally well represented in the surviving papyri. The development of any hand is largely influenced by

23393-414: Was applied. The inscription of Yajna Sri Satakarni (2nd century) from Amaravati is considered to be the earliest so far. The earlier writings (4th century) of Salankayanas of the Telugu region are in Prakrit, while their later records (belonging to the 5th century) are written in Sanskrit. In the Kannada speaking area , inscriptions belonging to later Satavahanas and Chutus were written in Prakrit. From

23550-457: Was at its height. By this time it had become an imitative hand, in which there was generally no room for spontaneous development. It remained noticeably uniform over a long period. It is difficult therefore to date the manuscripts by palaeographical criteria alone. The most that can be done is to classify them by centuries, on the strength of tenuous data. The earliest uncial writing is easily distinguished by its simple and monumental character from

23707-560: Was based on π = 6 arctan ⁡ ( 1 / 3 ) : {\displaystyle \pi =6\arctan(1/{\sqrt {3}}):} He used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.141 592 653 59 . He also improved the formula based on arctan(1) by including a correction: It is not known how he came up with this correction. Using this he found an approximation of π to 13 decimal places of accuracy when  n  = 75. Jamshīd al-Kāshī (Kāshānī),

23864-517: Was challenged by Falk and others in the early 1990s. In the new paradigm, Indian alphabetic writing, called Brahmi , was discontinuous with earlier, undeciphered, glyphs, and was invented specifically by King Ashoka for application in his royal edicts 250 BC . In the subcontinent, Kharosthi (clearly derived from the Aramaic alphabet ) was used at the same time in the northwest, next to Brahmi (at least influenced by Aramaic) elsewhere. In addition,

24021-482: Was made until the late medieval period. Some Egyptologists have claimed that the ancient Egyptians used an approximation of π as 22 ⁄ 7 = 3.142857 (about 0.04% too high) from as early as the Old Kingdom (c. 2700–2200 BC). This claim has been met with skepticism. Babylonian mathematics usually approximated π to 3, sufficient for the architectural projects of the time (notably also reflected in

24178-593: Was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer Madhava of Sangamagrama , founder of the Kerala school of astronomy and mathematics , found the Maclaurin series for arctangent, and then two infinite series for π . One of them is now known as the Madhava–Leibniz series , based on π = 4 arctan ⁡ ( 1 ) : {\displaystyle \pi =4\arctan(1):} The other

24335-438: Was said to have calculated new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of π until the advent of the electronic digital computer three-quarters of a century later. In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of π , including which computes a further eight decimal places of π with each term in

24492-407: Was sporadic in papyri, where they were used as an aid to understanding, and therefore more frequently in poetry than prose, and in lyrical oftener than in other verse. In the cursive of papyri they are practically unknown, as are marks of punctuation. Punctuation was effected in early papyri, literary and documentary, by spaces, reinforced in the book-hand by the paragraphos , a horizontal stroke under

24649-649: Was the ancestor of the modern Arabic and Hebrew scripts , as well as the Brahmi script , the parent writing system of most modern abugidas in India, Southeast Asia, Tibet, and Mongolia. Initially, the Aramaic script did not differ from the Phoenician, but then the Aramaeans simplified some of the letters, thickened and rounded their lines: a specific feature of its letters is the distinction between ⟨d⟩ and ⟨r⟩ . One innovation in Aramaic

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