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Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Modern usage employs seven symbols, each with a fixed integer value:^{[1]}
The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some minor applications to this day.
One place they are often seen is on clock faces. For instance, on the clock of Big Ben (designed in 1852), the hours from 1 to 12 are written as:
- I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII
The notations IV and IX can be read as "one less than five" (4) and "one less than ten" (9). On most Roman numeral clock faces, however, 4 is traditionally written as IIII.^{[2]}
Other common uses include year numbers on monuments and buildings and copyright dates on the title screens of movies and television programs. MCM, signifying "a thousand, and a hundred less than another thousand", means 1900, so 1912 is written MCMXII. For this century, MM indicates 2000. Thus the current year is MMXIX (2019).
Description
There is not, and never has been, an "official", "binding", or universally accepted standard for Roman numerals. Usage in ancient Rome varied greatly and remained somewhat inconsistent in medieval times and later.^{[3]} The "rules" of the system as it is now applied have been established only by general usage over the centuries.
A "base 10" system
Roman numerals are essentially a decimal or "base 10" number system. Powers of ten – thousands, hundreds, tens and units – are written separately, from left to right, in that order. Different symbols are used for each power of ten, but a common pattern is used for each of them.
The underlying form of this pattern employs the symbols I and V (representing 1 and 5) as simple tally marks, to build the numbers from 1 to 9. Each marker for 1 (I) adds a unit value up to 5 (V), and is then added to (V) to make the numbers from 6 to 9. Finally the unit symbol for the next power completes a "finger count" sequence:
- I, II, III, IIII, V, VI, VII, VIII, VIIII, X.
At some early time the Romans started to use the abbreviated forms IV ("one less than 5") and IX ("one less than 10") for IIII and VIIII, a convention that has been widely, although not universally, used ever since.^{[a]} This convention is called "subtractive" notation,^{[4]}as opposed to the purely "additive" notation of IIII and VIIII.^{[5]} Thus the numbers from 1 to 10 are generally written as
- I, II, III, IV, V, VI, VII, VIII, IX, X.^{[6]}
The multiples of 10, from 10 to 100, are written according to the same pattern, with X, L, and C taking the place of I, V, and X
- X, XX, XXX, XL, L, LX, LXX, LXXX, XC, C.
Note that 40 is usually written XL ("10 less than 50") rather than XXXX, and 90 as XC ("10 less than 100") rather than LXXXX.
Similarly, the multiples of 100, 100 to 1000, are written as
- C, CC, CCC, CD, D, DC, DCC, DCCC, CM, M.
where CD is to be read as "100 less than 500" (that is, 400), and CM as "100 less than 1000" (that is, 900).
Since the system has no standard symbols for 5,000 and 10,000, the full pattern cannot be extended to the multiples of 1000 – restricting the "thousands" range of "normal" Roman numerals to 1,000, 2,000 and 3,000:
- M, MM, MMM.
A number containing several decimal places is represented, as in the Arabic system, by writing its power-of-ten parts — thousands, hundreds, tens and units — in sequence, from left to right, in descending order of value. For example:
- 39 = 30 + 9 = XXX + IX = XXXIX.
- 246 = 200 + 40 + 6 = CC + XL + VI = CCXLVI.
- 789 = 700 + 80 + 9 = DCC + LXXX + IX = DCCLXXXIX.
- 2,421 = 2000 + 400 + 20 + 1 = MM + CD + XX + I = MMCDXXI.
Any missing place (represented by a zero in the Arabic equivalent) is omitted, as in Latin (and English) speech:
- 160 = 100 + 60 = C + LX = CLX
- 207 = 200 + 7 = CC + VII = CCVII
- 1,009 = 1,000 + 9 = M + IX = MIX
- 1,066 = 1,000 + 60 + 6 = M + LX + VI = MLXVI^{[7]}^{[8]}
Roman numerals for large numbers are nowadays seen mainly in the form of year numbers, as in these examples:
- 1776 = 1,000 + 700 + 70 + 6 = M + DCC + LXX + VI = MDCCLXXVI (the date written on the book held by the Statue of Liberty).^{[9]}
- 1954 = 1,000 + 900 + 50 + 4 = M + CM + L + IV = MCMLIV (as in the trailer for the movie The Last Time I Saw Paris)^{[10]}
- 2014 = 2,000 + 10 + 4 = MM + X + IV = MMXIV (the year of the games of the XXII (22nd) Olympic Winter Games (in Sochi)
- The current year (2019) is MMXIX.
The largest number that can be represented in this notation is 3,999 (3,000 + 900 + 90 + 9 = MMM + CM + XC + IX = MMMCMXCIX).^{[b]}
Use of additive notation
While subtractive notation for multiples of 4 (IV, XL and CD) has been prevalent since Roman times, additive notation (IIII, XXXX,^{[11]} and CCCC^{[11]}) continued to be used, including in compound numbers like XXIIII,^{[12]} LXXIIII,^{[13]} and CCCCLXXXX.^{[14]} The additive forms for 9, 90, and 900 (VIIII,^{[11]}LXXXX,^{[15]} and DCCCC^{[16]}) have also been used, although less frequently.
The two conventions could be mixed in the same document or inscription, even in the same numeral. On the numbered gates to the Colosseum, for instance, IIII is systematically used instead of IV, but subtractive notation is used for other digits; so that gate 44 is labelled XLIIII.^{[17]} Isaac Asimov speculates that the use of IV, as the initial letters of "IVPITER" (a classical Latin spelling of the name of the Roman god Jupiter) may have been felt to have been impious in this context.^{[18]}
Modern clock faces that use Roman numerals still usually employ IIII for four o'clock but IX for nine o'clock, a practice that goes back to very early clocks such as the Wells Cathedral clock of the late 14th century.^{[19]}^{[20]}^{[21]} However, this is far from universal: for example, the clock on the Palace of Westminster tower, "Big Ben", uses a subtractive IV for 4 o'clock.^{[20]}
Several monumental inscriptions created in the early 20th century use variant forms for "1900" (usually written MCM). These vary from MDCCCCX - a classical use of additive notation for MCMX (1910), as seen on Admiralty Arch, London, to the more unusual, if not unique MDCDIII for MCMIII (1903), on the north entrance to the Saint Louis Art Museum.^{[22]}
Sometimes 5 and 50 have been written IIIII and XXXXX instead of V and L, and there are instances such as IIIIII and XXXXXX rather than VI or LX.^{[23]}^{[24]}
Irregular subtractive notation
The irregular use of subtractive notation, such as IIIXX for 17,^{[25]} IIXX for 18,^{[26]} IIIC for 97,^{[27]} IIC for 98,^{[28]}^{[29]} and IC for 99^{[30]} were occasionally used in more modern times. A possible explanation is that the word for 18 in Latin was duodeviginti, literally "two from twenty". Similarly, the words for 98 and 99 were duodecentum (two from hundred) and undecentum (one from hundred), respectively.^{[31]} These ways of saying 18, 98 and 99 have been attributed to influence from the Etruscans, who would say ciem zaθrum (three from twenty) for 17, eslem zaθrum (two from twenty) for 18 and θunem zaθrum (one from twenty) for 19.^{[32]} Apparently, at least one ancient stonecutter mistakenly thought that the IIXX of "22nd Legion" stood for 18, and "corrected" it to XVIII.^{[33]} However, the explanation does not seem to apply to IIIXX and IIIC, since the Latin words for 17 and 97 were septendecim (seven ten) and nonaginta septem (ninety seven), respectively.
Another example of irregular subtractive notation is the use of XIIX for 18. It was used by officers of the XVIII Roman Legion to write their number.^{[34]}^{[35]} The notation appears prominently on the cenotaph of their senior centurion Marcus Caelius (c. 45 BC – AD 9). There does not seem to be a linguistic explanation for this use, although it is one stroke shorter than XVIII.
Rare variants
While the subtractive and additive notations seem to have been used interchangibly through history, some other Roman numerals have been occasionally observed that do not fit either system. Some of these variants do not seem to have been used outside specific contexts, and may have been regarded as errors even by contemporaries.
- IIXX was how people associated with the XXII Roman Legion used to write their number. The practice may have been due to a common way to say "twenty-second" in Latin, namely duo et vice(n)sima (literally "two and twentieth") rather than the "regular" vice(n)sima secunda (twentieth second).^{[33]}
- There are some examples of year numbers after 1000 written as two Roman numerals 1-99, e.g. 1613 as XVIXIII, corresponding to the common reading "sixteen thirteen" of such year numbers in English, or 1519 as XV^{C}XIX as in French quinze-cent-dix-neuf (fifteen-hundred-ten-nine), and similar readings in other languages.^{[37]}
- In some French texts from the 15th century and later one finds constructions like IIII^{XX}XIX for 99, reflecting the French reading of that number as quatre-vingt-dix-neuf (four-score and nineteen).^{[37]} Similarly, in some English documents one finds, for example, 77 written as "iii^{xx}xvii" (which could be read "three-score and seventeen").^{[38]}
- Another medieval accounting text from 1301 renders numbers like 13,573 as "XIII. M. V. C. III. XX. XIII, that is, "(13 × 1000) + (5 × 100) + (3 × 20) + 13".^{[39]}
- Other numerals that do not fit the usual patterns — such as VXL for 45, instead of the usual XLV — may be due to scribal errors, or the writer's lack of familiarity with the system, rather than being genuine variant usage.
- Not all combinations of symbols used in Roman numerals are intended to be taken numerically. For example "XXX" and "XL" have other connotations besides their values as Roman numerals, while "IXL" more often than not means "I excel", and is in any case not an unambiguous Roman numeral.
Origin of the system
The system as we use today is closely associated with the city of Rome and the Empire that it created. However, due to the scarcity of surviving examples, the origins of the system are obscure and there are several competing theories, largely conjectural.
Precedents
By the time the city-state of Rome was founded, the Mediterranean east of Italy had already known several advanced literate civilizations, such as the Ancient Egypt, Phoenicia, Minos, Mycenae and several others in Asia Minor and the Levant. These civilizations influenced the Romans directly or indirectly, such as in the alphabet (via the Etruscans).
The most obvious similarity between the Roman numerals and those older number systems is the use of 10 as the base, instead of 60 (as had been in use in Mesopotamia for a millennium) or 20. That choice, however, seems to have been made through most of Eurasia, including in India and China.
The old Egyptian system had a different non-phonetic symbol for each power of ten (so that, like the Romans, they did not need a symbol for zero). However, it repeated that symbol up to nine times to get the digits 1 to 9. The Egyptians had no separate symbols for 5, 50, etc. The symbol for 1 was a vertical stroke, and that for 10 was an inverted "U"; but higher powers were denoted by figurative hieroglyphs.
The use of separate symbols for 5, 50, 500 etc. was a feature of the old Greek number system, the Attic numerals. The additive notation for writing the digits 6-9 (as the digit 5 plus the digits 1-4) was also a feature of that system. However, the symbols of the latter were letters of the Greek alphabet taken from the names of those numbers ("Π" from ΠΕΝΤΕ for 5), "Δ" from ΔΕΚΑ for 10, "Η" from ΗΕΚΑΤΟΝ for 100, etc. or modified "Π" like "𐅄" for 50 an "𐅅" for 500.
The Attic numerals were later replaced in Greece by the classical Greek system, that also used letters to denote numbers, like the Romans, and a different set of letters for each power of 10. However, the similarities ended there. The Greek system (apparently borrowed from the late Egyptian Hieratic system) uses the first 9 letters of the alphabet to encode the units 1 to 9, then the next 9 letters for the tens 10 to 90, and similarly for the hundreds. (For this purpose an archaic alphabet was used which included additional letters not present in the classical Greek alphabet.)
Etruscan numerals
Rome was founded sometime between 850 and 750 BC. At the time, the region was inhabited by diverse populations speaking distinct languages. These included several Italic branches of the Indo-European family, as well as the Etruscans, a people of unknown origin who spoke an isolated language.
At that time, the Etruscans were the most advanced civilization in the region, and the ancient Romans themselves admitted that they inherited much of their knowledge and customs from them. Rome was located next to the southern edge of the Etruscan domain, which covered a large part of north-central Italy. Besides the Etruscans, the original population of Rome was allegedly drawn mainly from the Latins and the Sabinians, who were only two among the several Italic-speaking tribes in the peninsula.
The Roman numerals, in particular, are directly derived from the Etruscan number symbols: "𐌠", "𐌡", "𐌢", "𐌣", and "𐌟" for 1, 5, 10, 50, and 100 (They had more symbols for larger numbers, but it is unknown which symbol represents which number). As in the basic Roman system, the Etruscans wrote the symbols that added to the desired number, from higher to lower value. Thus the number 87, for example, would be written 50 + 10 + 10 + 10 + 5 + 1 + 1 = 𐌣𐌢𐌢𐌢𐌡𐌠𐌠 (this would appear as 𐌠𐌠𐌡𐌢𐌢𐌢𐌣 since Etruscan was written from right to left.)^{[40]}
The symbols "𐌠" and "𐌡" resembled letters of the Etruscan alphabet, but "𐌢", "𐌣", and "𐌟" did not. The Etruscans used the subtractive notation, too, but not like the Romans. They wrote 17, 18, and 19 as "𐌠𐌠𐌠𐌢𐌢", "𐌠𐌠𐌢𐌢", and 𐌠𐌢𐌢, mirroring the way they spoke those numbers ("three from twenty", etc.); and similarly for 27, 28, 29, 37, 38, etc. However they did not write "𐌠𐌡" for 4 (or "𐌢𐌣" for 40), and wrote "𐌡𐌠𐌠", "𐌡𐌠𐌠𐌠" and "𐌡𐌠𐌠𐌠𐌠" for 7, 8, and 9, respectively.^{[40]}
Early Roman numerals
The early Roman numerals for 1, 10, and 100 were the Etruscan ones: "I", "X", and "Ж". The symbols for 5 and 50 changed from Ʌ and "𐌣" to V and ↆ at some point. The latter had flattened to ⊥ (an inverted T) by the time of Augustus, and soon afterwards became identified with the graphically similar letter L.^{[41]}
The symbol for 100 was written variously as >I< or ƆIC, was then abbreviated to Ɔ or C, with C (which matched a Latin letter) finally winning out. It may have helped that C is the initial of centum, Latin for "hundred".^{[citation needed]}
The numbers 500 and 1000 were denoted by V or X overlaid with a box or circle. Thus 500 was like a Ɔ superimposed on a Þ. It became D or Ð by the time of Augustus, under the graphic influence of the letter D. It was later identified as the letter D; an alternative symbol for "thousand" was a CIƆ, and half of a thousand or "five hundred" is the right half of the symbol, IƆ, and this may have been converted into D.^{[18]}
The notation for 1000 was a circled or boxed X: Ⓧ, ⊗, ⊕, and by Augustinian times was partially identified with the Greek letter Φ phi. Over time, the symbol changed to Ψ and ↀ. The latter symbol further evolved into ∞, then ⋈, and eventually changed to M under the influence of the Latin word mille "thousand".^{[41]}
According to Paul Kayser, the basic numerical symbols were I, X, C and Φ (or ⊕) and the intermediate ones were derived by taking half of those (half an X is V, half a C is L and half a Φ/⊕ is D).^{[42]}
Use in the Middle Ages and Renaissance
Lower case, minuscule, letters were developed in the Middle Ages, well after the demise of the Western Roman Empire, and since that time lower-case versions of Roman numbers have also been commonly used: i, ii, iii, iv, and so on.
Since the Middle Ages, a "j" has sometimes been substituted for the final "i" of a "lower-case" Roman numeral, such as "iij" for 3 or "vij" for 7. This "j" can be considered a swash variant of "i". The use of a final "j" is still used in medical prescriptions to prevent tampering with or misinterpretation of a number after it is written.^{[43]}^{[44]}
Numerals in documents and inscriptions from the Middle Ages sometimes include additional symbols, which today are called "medieval Roman numerals". Some simply substitute another letter for the standard one (such as "A" for "V", or "Q" for "D"), while others serve as abbreviations for compound numerals ("O" for "XI", or "F" for "XL"). Although they are still listed today in some dictionaries, they are long out of use.^{[45]}
Number | Medieval abbreviation |
Notes and etymology |
---|---|---|
5 | A | Resembles an upside-down V. Also said to equal 500. |
6 | ↅ | Either from a ligature of VI, or from digamma (ϛ), the Greek numeral 6 (sometimes conflated with the stigma ligature).^{[41]} |
7 | S, Z | Presumed abbreviation of septem, Latin for 7. |
9.5 | X ̷ | Scribal abbreviation, an x with a slash through it. Likewise, IX ̷ represented 8.5 |
11 | O | Presumed abbreviation of onze, French for 11. |
40 | F | Presumed abbreviation of English forty. |
70 | S | Also could stand for 7, with the same derivation. |
80 | R | |
90 | N | Presumed abbreviation of nonaginta, Latin for 90. (Ambiguous with N for "nothing" (nihil)). |
150 | Y | Possibly derived from the lowercase y's shape. |
151 | K | Unusual, origin unknown; also said to stand for 250.^{[46]} |
160 | T | Possibly derived from Greek tetra, as 4 × 40 = 160. |
200 | H | Could also stand for 2 (see also 𐆙, the symbol for the dupondius). From a barring of two I's. |
250 | E | |
300 | B | |
400 | P, G | |
500 | Q | Redundant with D; abbreviates quingenti, Latin for 500. |
800 | Ω | Borrowed from Gothic. |
2000 | Z |
Chronograms, messages with dates encoded into them, were popular during the Renaissance era. The chronogram would be a phrase containing the letters I, V, X, L, C, D, and M. By putting these letters together, the reader would obtain a number, usually indicating a particular year.
Modern use
By the 11th century, Arabic numerals had been introduced into Europe from al-Andalus, by way of Arab traders and arithmetic treatises. Roman numerals, however, proved very persistent, remaining in common use in the West well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been made using an abacus). Replacement by their more convenient "Arabic" equivalents was quite gradual, and Roman numerals are still used today in certain contexts. A few examples of their current use are:
- Names of monarchs and popes, e.g. Elizabeth II of the United Kingdom, Pope Benedict XVI. These are referred to as regnal numbers and are usually read as ordinals; e.g. II is pronounced "the second". This tradition began in Europe sporadically in the Middle Ages, gaining widespread use in England during the reign of Henry VIII. Previously, the monarch was not known by numeral but by an epithet such as Edward the Confessor. Some monarchs (e.g. Charles IV of Spain and Louis XIV of France) seem to have preferred the use of IIII instead of IV on their coinage (see illustration).
- Generational suffixes, particularly in the US, for people sharing the same name across generations, for example William Howard Taft IV.
- In the French Republican Calendar, initiated during the French Revolution, years were numbered by Roman numerals – from the year I (1792) when this calendar was introduced to the year XIV (1805) when it was abandoned.
- The year of production of films, television shows and other works of art within the work itself. It has been suggested – by BBC News, perhaps facetiously – that this was originally done "in an attempt to disguise the age of films or television programmes."^{[47]} Outside reference to the work will use regular Arabic numerals.
- Hour marks on timepieces. In this context, 4 is often written IIII.
- The year of construction on building faces and cornerstones.
- Page numbering of prefaces and introductions of books, and sometimes of appendices and annexes, too.
- Book volume and chapter numbers, as well as the several acts within a play (e.g. Act iii, Scene 2).
- Sequels to some films, video games, and other works (as in Rocky II).
- Outlines that use numbers to show hierarchical relationships.
- Occurrences of a recurring grand event, for instance:
- The Summer and Winter Olympic Games (e.g. the XXI Olympic Winter Games; the Games of the XXX Olympiad)
- The Super Bowl, the annual championship game of the National Football League (e.g. Super Bowl XXXVII; Super Bowl 50 is a one-time exception^{[48]})
- WrestleMania, the annual professional wrestling event for the WWE (e.g. WrestleMania XXX). This usage has also been inconsistent.
Specific disciplines
In astronomy, the natural satellites or "moons" of the planets are traditionally designated by capital Roman numerals appended to the planet's name. For example, Titan's designation is Saturn VI.
In chemistry, Roman numerals are often used to denote the groups of the periodic table. They are also used in the IUPAC nomenclature of inorganic chemistry, for the oxidation number of cations which can take on several different positive charges. They are also used for naming phases of polymorphic crystals, such as ice.
In education, school grades (in the sense of year-groups rather than test scores) are sometimes referred to by a Roman numeral; for example, "grade IX" is sometimes seen for "grade 9".
In entomology, the broods of the thirteen and seventeen year periodical cicadas are identified by Roman numerals.
In advanced mathematics (including trigonometry, statistics, and calculus), when a graph includes negative numbers, its quadrants are named using I, II, III, and IV. These quadrant names signify positive numbers on both axes, negative numbers on the X axis, negative numbers on both axes, and negative numbers on the Y axis, respectively. The use of Roman numerals to designate quadrants avoids confusion, since Arabic numerals are used for the actual data represented in the graph.
In military unit designation, Roman numerals are often used to distinguish between units at different levels. This reduces possible confusion, especially when viewing operational or strategic level maps. In particular, army corps are often numbered using Roman numerals (for example the American XVIII Airborne Corps or the WW2-era German III Panzerkorps) with Arabic numerals being used for divisions and armies.
In music, Roman numerals are used in several contexts:
- Movements are often numbered using Roman numerals.
- In music theory, the diatonic functions are identified using Roman numerals. (See: Roman numeral analysis)
- Individual strings of stringed instruments, such as the violin, are often denoted by Roman numerals, with higher numbers denoting lower strings.
In pharmacy, Roman numerals are used in some contexts, including S to denote "one half" and N to denote "zero" (See the sections below on "zero" and "fractions").^{[49]}
In photography, Roman numerals (with zero) are used to denote varying levels of brightness when using the Zone System.
In seismology, Roman numerals are used to designate degrees of the Mercalli intensity scale of earthquakes.
In sport the team containing the "top" players and representing a nation or province, a club or a school at the highest level in (say) rugby union is often called the "1st XV", while a lower-ranking cricket or American football team might be the "3rd XI".
In tarot, Roman numerals (with zero) are used to denote the cards of the Major Arcana.
In theology and biblical scholarship, the Septuagint is often referred to as LXX, as this translation of the Old Testament into Greek is named for the legendary number of its translators (septuaginta being Latin for "seventy").
Modern use in continental Europe
Some uses that are rare or never seen in English speaking countries may be relatively common in parts of continental Europe. For instance:
Capital or small capital Roman numerals are widely used in Romance languages to denote centuries, e.g. the French xviii^{e} siècle^{[50]} and the Spanish siglo XVIII mean "18th century". Slavic languages in and adjacent to Russia similarly favour Roman numerals (XVIII век). On the other hand, in Slavic languages in Central Europe, like most Germanic languages, one writes "18." (with a period) before the local word for "century".
Mixed Roman and Arabic numerals are sometimes used in numeric representations of dates (especially in formal letters and official documents, but also on tombstones). The month is written in Roman numerals, while the day is in Arabic numerals: "14.VI.1789" and "VI.14.1789" both refer unambiguously to 14 June 1789.
Roman numerals are sometimes used to represent the days of the week in hours-of-operation signs displayed in windows or on doors of businesses,^{[51]} and also sometimes in railway and bus timetables. Monday, taken as the first day of the week, is represented by I. Sunday is represented by VII. The hours of operation signs are tables composed of two columns where the left column is the day of the week in Roman numerals and the right column is a range of hours of operation from starting time to closing time. In the example case (left), the business opens from 10 am to 7 pm on weekdays, 10 AM to 5 pm on Saturdays and is closed on Sundays. Note that the listing uses 24-hour time.
Roman numerals may also be used for floor numbering.^{[52]}^{[53]} For instance, apartments in central Amsterdam are indicated as 138-III, with both an Arabic numeral (number of the block or house) and a Roman numeral (floor number). The apartment on the ground floor is indicated as 138-huis.
In Italy, where roads outside built-up areas have kilometre signs, major roads and motorways also mark 100-metre subdivisionals, using Roman numerals from I to IX for the smaller intervals. The sign "IX | 17" thus marks 17.9 km.
A notable exception to the use of Roman numerals in Europe is in Greece, where Greek numerals (based on the Greek alphabet) are generally used in contexts where Roman numerals would be used elsewhere.
Special values
Zero
The number zero does not have its own Roman numeral, but the word nulla (the Latin word meaning "none") was used by medieval scholars in lieu of 0. Dionysius Exiguus was known to use nulla alongside Roman numerals in 525.^{[54]}^{[55]} About 725, Bede or one of his colleagues used the letter N, the initial of nulla or of nihil (the Latin word for "nothing"), in a table of epacts, all written in Roman numerals.^{[56]}
Fractions
Though the Romans used a decimal system for whole numbers, reflecting how they counted in Latin, they used a duodecimal system for fractions, because the divisibility of twelve (12 = 2^{2} × 3) makes it easier to handle the common fractions of 1/3 and 1/4 than does a system based on ten (10 = 2 × 5). On coins, many of which had values that were duodecimal fractions of the unit as, they used a tally-like notational system based on twelfths and halves. A dot (·) indicated an uncia "twelfth", the source of the English words inch and ounce; dots were repeated for fractions up to five twelfths. Six twelfths (one half) was abbreviated as the letter S for semis "half". Uncia dots were added to S for fractions from seven to eleven twelfths, just as tallies were added to V for whole numbers from six to nine.^{[57]}
Each fraction from 1/12 to 12/12 had a name in Roman times; these corresponded to the names of the related coins:
Fraction | Roman numeral | Name (nominative and genitive) | Meaning |
---|---|---|---|
1/12 | · | Uncia, unciae | "Ounce" |
2/12 = 1/6 | ·· or : | Sextans, sextantis | "Sixth" |
3/12 = 1/4 | ··· or ∴ | Quadrans, quadrantis | "Quarter" |
4/12 = 1/3 | ···· or ∷ | Triens, trientis | "Third" |
5/12 | ····· or ⁙ | Quincunx, quincuncis | "Five-ounce" (quinque unciae → quincunx) |
6/12 = 1/2 | S | Semis, semissis | "Half" |
7/12 | S· | Septunx, septuncis | "Seven-ounce" (septem unciae → septunx) |
8/12 = 2/3 | S·· or S: | Bes, bessis | "Twice" (as in "twice a third") |
9/12 = 3/4 | S··· or S∴ | Dodrans, dodrantis or nonuncium, nonuncii |
"Less a quarter" (de-quadrans → dodrans) or "ninth ounce" (nona uncia → nonuncium) |
10/12 = 5/6 | S···· or S∷ | Dextans, dextantis or decunx, decuncis |
"Less a sixth" (de-sextans → dextans) or "ten ounces" (decem unciae → decunx) |
11/12 | S····· or S⁙ | Deunx, deuncis | "Less an ounce" (de-uncia → deunx) |
12/12 = 1 | I | As, assis | "Unit" |
The arrangement of the dots was variable and not necessarily linear. Five dots arranged like (⁙) (as on the face of a die) are known as a quincunx, from the name of the Roman fraction/coin. The Latin words sextans and quadrans are the source of the English words sextant and quadrant.
Other Roman fractional notations included the following:
- 1/8 sescuncia, sescunciae (from sesqui- + uncia, i.e. 1^{1}⁄_{2} uncias), represented by a sequence of the symbols for the semuncia and the uncia.
- 1/24 semuncia, semunciae (from semi- + uncia, i.e. ^{1}⁄_{2} uncia), represented by several variant glyphs deriving from the shape of the Greek letter sigma (Σ), one variant resembling the pound sign without the horizontal line (𐆒) and another resembling the Cyrillic letter Є.
- 1/36 binae sextulae, binarum sextularum ("two sextulas") or duella, duellae, represented by a sequence of two reversed Ss (ƧƧ).
- 1/48 sicilicus, sicilici, represented by a reversed C (Ɔ).
- 1/72 sextula, sextulae (^{1}⁄_{6} of an uncia), represented by a reversed S (Ƨ).
- 1/144 = 12^{−2} dimidia sextula, dimidiae sextulae ("half a sextula"), represented by a reversed S crossed by a horizontal line (
Ƨ). - 1/288 scripulum, scripuli (a scruple), represented by the symbol ℈.
- 1/1728 = 12^{−3} siliqua, siliquae, represented by a symbol resembling closing guillemets (»).
Large numbers
A number of systems were developed for the expression of larger numbers that cannot be conveniently expressed using the normal seven letter symbols of conventional Roman numerals.
Apostrophus
One of these was the apostrophus,^{[58]} in which 500 (usually written as "D") was written as IƆ, while 1,000, was written as CIƆ instead of "M".^{[18]} This is a system of encasing numbers to denote thousands (imagine the Cs and Ɔs as parentheses), which has its origins in Etruscan numeral usage. The IƆ and CIƆ used to represent 500 and 1,000 most likely preceded, and subsequently influenced, the adoption of "D" and "M" in conventional Roman numerals.
In this system, an extra Ɔ denoted 500, and multiple extra Ɔs are used to denote 5,000, 50,000, etc. For example:
Base number | CIƆ = 1,000 | CCIƆƆ = 10,000 | CCCIƆƆƆ = 100,000 | |
---|---|---|---|---|
1 extra Ɔ | IƆ = 500 | CIƆƆ = 1,500 | CCIƆƆƆ = 10,500 | CCCIƆƆƆƆ = 100,500 |
2 extra Ɔs | IƆƆ = 5,000 | CCIƆƆƆƆ = 15,000 | CCCIƆƆƆƆƆ = 105,000 | |
3 extra Ɔs | IƆƆƆ = 50,000 | CCCIƆƆƆƆƆƆ = 150,000 |
Sometimes CIƆ was reduced to ↀ for 1,000. John Wallis is often credited for introducing the symbol for infinity (modern ∞), and one conjecture is that he based it on this usage, since 1,000 was hyperbolically used to represent very large numbers. Similarly, IƆƆ for 5,000 was reduced to ↁ; CCIƆƆ for 10,000 to ↂ; IƆƆƆ for 50,000 to ↇ; and CCCIƆƆƆ for 100,000 to ↈ.^{[59]}
Vinculum
Another system is the vinculum, in which conventional Roman numerals are multiplied by 1,000 by adding a "bar" or "overline".^{[59]} Although mathematical historian David Eugene Smith disputes that this was part of ancient Roman usage,^{[60]} the notation was certainly in use in the Middle Ages, and is sometimes suggested as a workable method for modern use, although it is not standardised as such.
Any hundreds, tens or units in the number are written in ordinary Roman numerals - but instead of M, MM or MMM, "barred" notation is used to express the thousands - which greatly expands the range of numbers expressible.
For instance:
- IV = 4,000
- IVDCXXVII = 4,627
- XXV = 25,000
- XXVCDLIX = 25,459
If this were ever to be applied consistently in our own times - then the main difficulty would be what to do with "M" - one way would be to do away with "M" altogether, except perhaps for CM (=900) - thus rendering MMXVIII as IIXVIII - or alternatively to retain "M" in its current usage, with the barred numerals starting at IV (=4,000). Retaining "M" would permit our numerals to run up to MMMCMXCIXCMXCIX (= 3,999,999).
Another inconsistent medieval usage was the addition of vertical lines (or brackets) before and after the numeral to multiply it by 10 (or 100): thus M for 10,000 as an alternative form for X. In combination with the overline the bracketed forms might be used to raise the multiplier to (say) ten (or one hundred) thousand, thus:
- VIII for 80,000 (or 800,000)
- XX for 200,000 (or 2,000,000)
Through all this, and whether any kind of vinculum notation or "barring" ought to be revived or not, this needs to be distinguished from the custom, once very common, of adding both underline and overline to a Roman numeral, simply to make it clear that it is a number, e.g. MCMLXVII (1967).
Unicode
The "Number Forms" block of the Unicode computer character set standard has a number of Roman numeral symbols in the range of code points from U+2160 to U+2188.^{[61]} This range includes both upper- and lowercase numerals, as well as pre-combined characters for numbers up to 12 (Ⅻ or XII). One justification for the existence of pre-combined numbers is to facilitate the setting of multiple-letter numbers (such as VIII) on a single horizontal line in Asian vertical text. The Unicode standard, however, includes special Roman numeral code points for compatibility only, stating that "[f]or most purposes, it is preferable to compose the Roman numerals from sequences of the appropriate Latin letters".^{[62]} The block also includes some apostrophus symbols for large numbers, an old variant of "L" (50) similar to the Etruscan character, the Claudian letter "reversed C", etc.
Symbol | ↀ | ↁ | ↂ | ↅ | ↆ | ↇ | ↈ |
---|---|---|---|---|---|---|---|
Value | 1,000 | 5,000 | 10,000 | 6 | 50 | 50,000 | 100,000 |
See also
- Egyptian numerals
- Etruscan numerals
- Greek numerals
- Kharosthi numerals
- Roman abacus
- Proto-writing
- Roman numerals in Unicode
- Pentimal system
References
Notes
- ^ Without theorising about causation, it may be noted that "IV" uses 25% fewer strokes than "IIII", and takes up 25% less space. For the other forms (IX, XL, XC, CD, and CM), the savings on either or both counts are 50% or more. "IX" is also more distinctive than "VIIII" and less likely to be confused with "VIII". This equally applies to XC, CD, and CM.
- ^ Since the largest Roman numerals likely to be used today are year numbers up to the present there is normally no need to use Roman numerals for numbers beyond this limit. In the unlikely case a larger number might be needed there is really no reason why more "M"s, as required, could not be added, cumbersome as this might prove. Through the centuries during which Roman numerals remained the standard way of writing numbers throughout Europe there were extensions to the system designed to indicate larger numbers, see the final sections of this article.
Citations
- ^ Gordon, Arthur E. (1982). Illustrated Introduction to Latin Epigraphy. Berkeley: University of California Press. ISBN 0-520-05079-7.
Alphabetic symbols for larger numbers, such as Q for 500,000, have also been used to various degrees of standardization.
- ^ "The Mathematical Tourist : IIII versus IV on Clocks". Ivars Peterson. Retrieved 31 May 2019.
Expressed as Roman numerals, the first twelve numbers are usually given as I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII. However, on many clock faces, when the numbers on the dial are in Roman numerals, IIII replaces IV.
- ^ Adams, Cecil (23 February 1990). "What is the proper way to style Roman numerals for the 1990s?". The Straight Dope.
- ^ Stanislas Dehaene (1997): The Number Sense : How the Mind Creates Mathematics. Oxford University Press; 288 pages. ISBN 9780199723096
- ^ Ûrij Vasilʹevič Prokhorov and Michiel Hazewinkel, editors (1990): Encyclopaedia of Mathematics, Volume 10, page 502. Springer; 546 pages. ISBN 9781556080050
- ^ Reddy, Indra K.; Khan, Mansoor A. (2003). Essential Math and Calculations for Pharmacy Technicians. CRC Press. ISBN 978-0-203-49534-6.
- ^ Dela Cruz, M. L. P.; Torres, H. D. (2009). Number Smart Quest for Mastery: Teacher's Edition. Rex Bookstore, Inc. ISBN 9789712352164.
- ^ Martelli, Alex; Ascher, David (2002). Python Cookbook. O'Reilly Media Inc. ISBN 978-0-596-00167-4.
- ^ "What book is the Statue of Liberty holding? What is its significance?". Quora.
- ^ Hayes, David P. "Guide to Roman Numerals". Copyright Registration and Renewal Information Chart and Web Site.
- ^ ^{a} ^{b} ^{c} Julius Caesar (52-49 BC): Commentarii de Bello Gallico. Book II, Section 4: "... XV milia Atrebates, Ambianos X milia, Morinos XXV milia, Menapios VII milia, Caletos X milia, Veliocasses et Viromanduos totidem, Atuatucos XVIIII milia; ..." Section 8: "... ab utroque latere eius collis transversam fossam obduxit circiter passuum CCCC et ad extremas fossas castella constituit..." Book IV, Section 15: "Nostri ad unum omnes incolumes, perpaucis vulneratis, ex tanti belli timore, cum hostium numerus capitum CCCCXXX milium fuisset, se in castra receperunt." Book VII, Section 4: "...in hiberna remissis ipse se recipit die XXXX Bibracte."
- ^ Angelo Rocca (1612) De campanis commentarius. Published by Guillelmo Faciotti, Rome. Title of a Plate: "Campana a XXIIII hominibus pulsata" ("Bell to be sounded by 24 men")
- ^ Gerard Ter Borch (1673): Portrait of Cornelis de Graef. Date on painting: "Out. XXIIII Jaer. // M. DC. LXXIIII".
- ^ Pliny the Elder (77-79 AD): Naturalis Historia, Book III: "Saturni vocatur, Caesaream Mauretaniae urbem CCLXXXXVII p[assum]. traiectus. reliqua in ora flumen Tader ... ortus in Cantabris haut procul oppido Iuliobrica, per CCCCL p. fluens ..." Book IV: "Epiri, Achaiae, Atticae, Thessalia in porrectum longitudo CCCCLXXXX traditur, latitudo CCLXXXXVII." Book VI: "tam vicinum Arsaniae fluere eum in regione Arrhene Claudius Caesar auctor est, ut, cum intumuere, confluant nec tamen misceantur leviorque Arsanias innatet MMMM ferme spatio, mox divisus in Euphraten mergatur."
- ^ Thomas Bennet (1731): Grammatica Hebræa, cum uberrima praxi in usum tironum ... Editio tertia. Published by T. Astley, copy in the British Library; 149 pages. Page 24: "PRÆFIXA duo sunt viz. He emphaticum vel relativum (de quo Cap VI Reg. LXXXX.) & Shin cum Segal sequente Dagesh, quod denotat pronomen relativum..."
- ^ Pico Della Mirandola (1486)`Conclusiones sive Theses DCCCC ("Conclusions, or 900 Theses").
- ^ "360:12 tables, 24 chairs, and plenty of chalk". Roman Numerals…not quite so simple.
- ^ ^{a} ^{b} ^{c} Asimov, Isaac (1966). Asimov On Numbers (PDF). Pocket Books, a division of Simon & Schuster, Inc. p. 12.
- ^ Milham, W.I. (1947). Time & Timekeepers. New York: Macmillan. p. 196.
- ^ ^{a} ^{b} Pickover, Clifford A. (2003), Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, Oxford University Press, p. 282, ISBN 978-0-19-534800-2.
- ^ Adams, Cecil; Zotti, Ed (1988). More of the straight dope. Ballantine Books. p. 154. ISBN 978-0-345-35145-6..
- ^ "Gallery: Museum's North Entrance (1910)". Saint Louis Art Museum. Archived from the original on 4 December 2010. Retrieved 10 January 2014.
The inscription over the North Entrance to the Museum reads: "Dedicated to Art and Free to All MDCDIII." These roman numerals translate to 1903, indicating that the engraving was part of the original building designed for the 1904 World's Fair.
- ^ Reynolds, Joyce Maire; Spawforth, Anthony J. S. (1996). "numbers, Roman". In Hornblower, Simon; Spawforth, Anthony (eds.). Oxford Classical Dictionary (3rd ed.). Oxford University Press. ISBN 0-19-866172-X.
- ^ Kennedy, Benjamin Hall (1923). The Revised Latin Primer. London: Longmans, Green & Co.
- ^ Michaele Gasp. Lvndorphio (1621): Acta publica inter invictissimos gloriosissimosque&c. ... et Ferdinandum II. Romanorum Imperatores.... Printed by Ian-Friderici Weissii. Page 123: "Sub Dato Pragæ IIIXX Decemb. A. C. M. DC. IIXX". Page 126, end of the same document: "Dabantur Pragæ 17 Decemb. M. DC. IIXX"
- ^ Raphael Sulpicius à Munscrod (1621): Vera Ac Germana Detecto Clandestinarvm Deliberationvm. Page 16, line 1: "repertum Originale Subdatum IIIXXX Aug. A. C. MDC.IIXX". Page 41, upper right corner: "Decemb. A. C. MDC.IIXX". Page 42, upper left corner: "Febr. A. C. MDC.XIX". Page 70: "IIXX. die Maij sequentia in consilio noua ex Bohemia allata....". Page 71: "XIX. Maij".
- ^ Wilhelm Ernst Tentzel (1699): Als Ihre Königl. Majestät in Pohlen und .... Page 39: "... und der Umschrifft: LITHUANIA ASSERTA M. DC. IIIC [1699]."
- ^ Joh. Caspar Posner (1698): Mvndvs ante mvndvm sive De Chao Orbis Primordio, title page: "Ad diem jvlii A. O. R. M DC IIC".
- ^ Wilhelm Ernst Tentzel (1700): Saxonia Nvmismatica: Das ist: Die Historie Des Durchlauchtigsten.... Page 26: "Die Revers hat eine feine Inscription: SERENISSIMO DN.DN... SENATUS.QVERNF. A. M DC IIC D. 18 OCT [year 1698 day 18 oct]."
- ^ Enea Silvio Piccolomini (1698): Opera Geographica et Historica. Helmstadt, J. M. Sustermann. Title page of first edition: "Bibliopolæ ibid. M DC IC"
- ^ Kennedy, Benjamin H. (1879). Latin grammar. London: Longmans, Green, and Co. p. 150.
- ^ Giuliano Bonfante (1985): "Etruscan Words in Latin". Word, volume 36, issue 3, pages 203-210. doi:10.1080/00437956.1985.11435872
- ^ ^{a} ^{b} Stephen James Malone, (2005) Legio XX Valeria Victrix.... PhD thesis. On page 396 it discusses many coins with "Leg. IIXX" and notes that it must be Legion 22. The footnote on that page says: "The form IIXX clearly reflecting the Latin duo et vicensima 'twenty-second': cf. X5398, legatus I[eg II] I et vicensim(ae) Pri[mi]g; VI 1551, legatus leg] IIXX Prj; III 14207.7, miles leg IIXX; and III 10471-3, a vexillation drawn from four German legions including 'XVIII PR' - surely here the stonecutter's hypercorrection for IIXX PR.
- ^ Adkins, Lesley; Adkins, Roy A (2004). Handbook to life in ancient Rome (2 ed.). p. 270. ISBN 0-8160-5026-0.
- ^ Boyne, William (1968). A manual of Roman coins. p. 13.
- ^ L' Atre périlleux et Yvain, le chevalier au lion . 1301–1350.CS1 maint: Date format (link)
- ^ ^{a} ^{b} M. Gachard (1862): "II. Analectes historiques, neuvième série (nos CCLXI-CCLXXXIV)". Bulletin de la Commission royale d'Historie, volume 3, pages 345-554. Page 347: Lettre de Philippe le Beau aux échevins..., quote: "Escript en nostre ville de Gand, le XXIIII^{me} de febvrier, l'an IIII^{XX}XIX [quatre-vingt-dix-neuf = 99]." Page 356: Lettre de l'achiduchesse Marguerite au conseil de Brabant..., quote: "... Escript à Bruxelles, le dernier jour de juing anno XV^{c}XIX [1519]." Page 374: Letters patentes de la rémission ... de la ville de Bruxelles, quote: "... Op heden, tweentwintich ['twenty-two'] daegen in decembri, anno vyfthien hondert tweendertich ['fifteen hundred thirty-two'] ... Gegeven op ten vyfsten dach in deser jegewoirdige maent van decembri anno XV tweendertich [1532] vorschreven." Page 419: Acte du duc de Parme portant approbation..., quote": "Faiet le XV^{me} de juillet XV^{c} huytante-six [1586]." doi:10.3406/bcrh.1862.3033
- ^ Herbert Edward Salter (1923) Registrum Annalium Collegii Mertonensis 1483-1521 Oxford Historical Society, volume 76; 544 pages. Page 184 has the computation in pounds:shillings:pence (li:s:d) x:iii:iiii + xxi:viii:viii + xlv:xiiii:i = iii^{xx}xvii:vi:i, i.e. 10:3:4 + 21:8:8 + 45:14:1 = 77:6:1.
- ^ Johannis de Sancto Justo (1301): "E Duo Codicibus Ceratis" ("From Two Texts in Wax"). In de Wailly, Delisle (1865): Contenant la deuxieme livraison des monumens des regnes de saint Louis,... Volume 22 of Recueil des historiens des Gaules et de la France. Page 530: "SUMMA totalis, XIII. M. V. C. III. XX. XIII. l. III s. XI d. [Sum total, 13 thousand 5 hundred 3 score 13 livres, 3 sous, 11 deniers].
- ^ ^{a} ^{b} Gilles Van Heems (2009)> "Nombre, chiffre, lettre : Formes et réformes. Des notations chiffrées de l'étrusque" ("Between Numbers and Letters: About Etruscan Notations of Numeral Sequences"). Revue de philologie, de littérature et d'histoire anciennes, volume LXXXIII (83), issue 1, pages 103-130. ISSN 0035-1652
- ^ ^{a} ^{b} ^{c} Perry, David J. Proposal to Add Additional Ancient Roman Characters to UCS Archived 22 June 2011 at the Wayback Machine.
- ^ Keyser, Paul (1988). "The Origin of the Latin Numerals 1 to 1000". American Journal of Archaeology. 92: 529–546. JSTOR 505248.
- ^ Sturmer, Julius W. Course in Pharmaceutical and Chemical Arithmetic, 3rd ed. (LaFayette, IN: Burt-Terry-Wilson, 1906). p25. Retrieved 15 March 2010.
- ^ Bastedo, Walter A. Materia Medica: Pharmacology, Therapeutics and Prescription Writing for Students and Practitioners, 2nd ed. (Philadelphia, PA: W.B. Saunders, 1919) p582. Retrieved 15 March 2010.
- ^ Capelli, A. Dictionary of Latin Abbreviations. 1912.
- ^ Bang, Jørgen. Fremmedordbog, Berlingske Ordbøger, 1962 (Danish)
- ^ Owen, Rob (13 January 2012). "TV Q&A: ABC News, 'Storage Wars' and 'The Big Bang Theory'". Pittsburgh Post-Gazette. Retrieved 13 January 2012.
- ^ NFL won't use Roman numerals for Super Bowl 50 Archived 1 December 2015 at the Wayback Machine, National Football League. Retrieved 5 November 2014
- ^ Bachenheimer, Bonnie S. (2010). Manual for Pharmacy Technicians. ISBN 1-58528-307-X.
- ^ Lexique des règles typographiques en usage à l'imprimerie nationale (in French) (6th ed.). Paris: Imprimerie nationale. March 2011. p. 126. ISBN 978-2-7433-0482-9. On composera en chiffres romains petites capitales les nombres concernant : ↲ 1. Les siècles.
- ^ Beginners latin Archived 3 December 2013 at the Wayback Machine, Government of the United Kingdom. Retrieved 1 December 2013
- ^ Roman Arithmetic Archived 22 November 2013 at the Wayback Machine, Southwestern Adventist University. Retrieved 1 December 2013
- ^ Roman Numerals History Archived 3 December 2013 at the Wayback Machine. Retrieved 1 December 2013
- ^ Faith Wallis, trans. Bede: The Reckoning of Time (725), Liverpool, Liverpool Univ. Pr., 2004. ISBN 0-85323-693-3.
- ^ Byrhtferth's Enchiridion (1016). Edited by Peter S. Baker and Michael Lapidge. Early English Text Society 1995. ISBN 978-0-19-722416-8.
- ^ C. W. Jones, ed., Opera Didascalica, vol. 123C in Corpus Christianorum, Series Latina.
- ^ Maher, David W.; Makowski, John F., "Literary Evidence for Roman Arithmetic with Fractions Archived August 27, 2013, at the Wayback Machine", Classical Philology 96 (2011): 376–399.
- ^ "Merriam-Webster Unabridged Dictionary".
- ^ ^{a} ^{b} Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos, E. F. Harding, Sophie Wood, Ian Monk. John Wiley & Sons.
- ^ Smith, David Eugene (1958) [1925], History of Mathematics, II, p. 60, ISBN 0-486-20430-8
- ^ Unicode Number Forms
- ^ The Unicode Standard, Version 6.0 – Electronic edition (PDF), Unicode, Inc., 2011, p. 486
Sources
- Menninger, Karl (1992). Number Words and Number Symbols: A Cultural History of Numbers. Dover Publications. ISBN 978-0-486-27096-8.
Further reading
- Aczel, Amir D. 2015. Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers. 1st edition. New York: Palgrave Macmillan.
- Goines, David Lance. A Constructed Roman Alphabet: A Geometric Analysis of the Greek and Roman Capitals and of the Arabic Numerals. Boston: D.R. Godine, 1982.
- Houston, Stephen D. 2012. The Shape of Script: How and Why Writing Systems Change. Santa Fe, NM: School for Advanced Research Press.
- Taisbak, Christian M. 1965. "Roman numerals and the abacus." Classica et medievalia 26: 147–60.
External links
Library resources about Roman numerals |
- "Roman Numerals (Totally Epic Guide)". Know The Romans.
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Rizom - April,2018For our ancestors who didn’t have the benefit of a world map...
Rizom - April,2018For our ancestors who didn’t have the benefit of a world map...
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