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43-419: In metal type, the point size of the font describes the height of the metal body on which the typeface 's characters were cast. In digital type, letters of a font are designed around an imaginary space called an em square . When a point size of a font is specified, the font is scaled so that its em square has a side length of that particular length in points. Although the letters of a font usually fit within

86-462: A proper divisor or an aliquot part of n {\displaystyle n} (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide n {\displaystyle n} but leaves a remainder is sometimes called an aliquant part of n . {\displaystyle n.} An integer n > 1 {\displaystyle n>1} whose only proper divisor

129-478: A bit larger than English points at around 0.350 mm . The Truchet point , the first modern typographic point, was 1 ⁄ 144 of a French inch or 1 ⁄ 1728 of the royal foot . It was invented by the French clergyman Sébastien Truchet . During the metrication of France amid its revolution , a 1799 law declared the meter to be exactly 443.296  French lines long. This established

172-510: A factor of √2 ≈ 1.414 in order to match ISO 216 paper sizes . Since the set of sizes includes thicknesses of 0.1 mm, 0.5 mm, 1 mm and 2 mm, there is also one of 0.35 mm which is almost exactly 1 pica point. In other words, 2 mm = 1 ⁄ √8 mm approximates an English typographic point rather well. The basic unit of measurements in American typography was the pica, usually approximated as one sixth of an inch, but

215-413: A length to the royal foot of 9000 ⁄ 27 706  m or about 325 mm. The Truchet point therefore became equal to 15 625 ⁄ 83 118  mm or about 0.187 986  mm . It has also been cited as exactly 0.188 mm. The Fournier point was established by Pierre Simon Fournier in 1737. The system of Fournier was based on a different French foot of c. 298 mm. With

258-565: A metric typographic base measure of exactly 1 ⁄ 4  mm or 0.250 mm , which is sometimes referred to as the quart in Japan. The symbol Q is used in Japanese after the initial letter of quarter millimeter . Due to demand by Japanese typesetters, CSS adopted Q in 2015. ISO 128 specifies preferred line thicknesses for technical drawings and ISO 9175 specifies respective pens. The steps between nominal sizes are based on

301-417: A number from the number's digits. There are some elementary rules: If a ∣ b c , {\displaystyle a\mid bc,} and gcd ( a , b ) = 1 , {\displaystyle \gcd(a,b)=1,} then a ∣ c . {\displaystyle a\mid c.} This is called Euclid's lemma . If p {\displaystyle p}

344-711: A type-foundry. Around 1790, Bache published a specimen sheet with some Fournier types. After the death of Franklin, the matrices and the Fournier mould were acquired by Binny and Ronaldson, the first permanent type-foundry in America. Successive mergers and acquisitions in 1833, 1860 and 1897 saw the company eventually become known as MacKellar, Smith & Jordan. The Fournier cicero mould was used by them to cast pica-sized type. Nelson Hawks proposed, like Fournier, to divide one American inch exactly into six picas, and one pica into 12 points. However, this saw an opposition because

387-540: A whole number of physical pixels in order to accommodate for screen size, pixel density and typical viewing distance. This Cocoa point is equivalent to the pixel px unit in CSS , the density-independent pixel dp on Android and the effective pixel epx or ep in Windows UWP . In lead typecasting, most font sizes commonly used in printing have conventional names that differ by country, language and

430-409: Is Euler–Mascheroni constant . One interpretation of this result is that a randomly chosen positive integer n has an average number of divisors of about ln ⁡ n . {\displaystyle \ln n.} However, this is a result from the contributions of numbers with "abnormally many" divisors . In definitions that allow the divisor to be 0, the relation of divisibility turns

473-427: Is 1 is called a prime number . Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself. Any positive divisor of n {\displaystyle n} is a product of prime divisors of n {\displaystyle n} raised to some power. This is a consequence of the fundamental theorem of arithmetic . A number n {\displaystyle n}

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516-499: Is a divisor of n {\displaystyle n} ; this implies dividing n {\displaystyle n} by m {\displaystyle m} leaves no remainder. An integer n {\displaystyle n} is divisible by a nonzero integer m {\displaystyle m} if there exists an integer k {\displaystyle k} such that n = k m . {\displaystyle n=km.} This

559-478: Is a multiple of m . {\displaystyle m.} If m {\displaystyle m} does not divide n , {\displaystyle n,} then the notation is m ∤ n . {\displaystyle m\not \mid n.} There are two conventions, distinguished by whether m {\displaystyle m} is permitted to be zero: Divisors can be negative as well as positive, although often

602-400: Is a prime number and p ∣ a b {\displaystyle p\mid ab} then p ∣ a {\displaystyle p\mid a} or p ∣ b . {\displaystyle p\mid b.} A positive divisor of n {\displaystyle n} that is different from n {\displaystyle n} is called

645-516: Is an integer m {\displaystyle m} that may be multiplied by some integer to produce n . {\displaystyle n.} In this case, one also says that n {\displaystyle n} is a multiple of m . {\displaystyle m.} An integer n {\displaystyle n} is divisible or evenly divisible by another integer m {\displaystyle m} if m {\displaystyle m}

688-441: Is defined as 1 ⁄ 72 or 0.013 8 of the international inch, making it equivalent to 25.4 ⁄ 72  mm = 0.352 7  mm. Twelve points make up a pica, and six picas make an inch. This specification was developed by John Warnock and Charles Geschke when they created Adobe PostScript . It was adopted by Apple Computer as the standard for the display resolution of the original Macintosh desktop computer and

731-1036: Is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than n , {\displaystyle n,} and abundant if this sum exceeds n . {\displaystyle n.} The total number of positive divisors of n {\displaystyle n} is a multiplicative function d ( n ) , {\displaystyle d(n),} meaning that when two numbers m {\displaystyle m} and n {\displaystyle n} are relatively prime , then d ( m n ) = d ( m ) × d ( n ) . {\displaystyle d(mn)=d(m)\times d(n).} For instance, d ( 42 ) = 8 = 2 × 2 × 2 = d ( 2 ) × d ( 3 ) × d ( 7 ) {\displaystyle d(42)=8=2\times 2\times 2=d(2)\times d(3)\times d(7)} ;

774-455: Is the first based on logical measurement called "Punto," which corresponds to the ninth part of the height of the letters or the thickness of the principal stroke . A measurement in points can be represented in three different ways. For example, 14 points (1 pica plus 2 points) can be written: There have been many definitions of a "point" since the advent of typography. Traditional continental European points at about 0.375 mm are usually

817-414: Is written as This may be read as that m {\displaystyle m} divides n , {\displaystyle n,} m {\displaystyle m} is a divisor of n , {\displaystyle n,} m {\displaystyle m} is a factor of n , {\displaystyle n,} or n {\displaystyle n}

860-680: The Monotype Corporation in 1927. It was still a standard in Belgium , in parts of Austria, and in Northern France at the beginning of the 20th century. In Belgium, the Fournier system was used until the 1970s and later. It was called the "mediaan"-system. The Didot point , established by François-Ambroise Didot in 1783, was an attempt to improve the Fournier system. He did not change the subdivisions (1 inch = 12 subdivisions = 72 points), but defined it strictly in terms of

903-415: The ascender and descender to allow additional space between the lines of text. More space might be achieved by inserting thin long pieces of lead between the lines of text (that is leading ). In digital fonts, the body is now a virtual, imaginary area, whose height still equals the point size as it did in metal type. The distance between one baseline and the next is the sum of the body height and

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946-406: The body height or point size refers to the height of the space in which a glyph is defined. Originally, in metal typesetting , the body height or the font (or point ) size was defined by the height of the lead cuboid ( metal sort ) on which the actual font face is moulded. The body height of a metal sort defined the point size , and was usually slightly larger than the distance between

989-438: The leading , often expressed as "characters per inch vertically" (as in RFC   678 ) or lines of text per inch (not to be confused with lines per inch , a measure of printed photograph resolution). Divisor In mathematics , a divisor of an integer n , {\displaystyle n,} also called a factor of n , {\displaystyle n,}

1032-737: The prime factorization of n {\displaystyle n} is given by then the number of positive divisors of n {\displaystyle n} is and each of the divisors has the form where 0 ≤ μ i ≤ ν i {\displaystyle 0\leq \mu _{i}\leq \nu _{i}} for each 1 ≤ i ≤ k . {\displaystyle 1\leq i\leq k.} For every natural n , {\displaystyle n,} d ( n ) < 2 n . {\displaystyle d(n)<2{\sqrt {n}}.} Also, where γ {\displaystyle \gamma }

1075-474: The prime factorization of 3 × 7 × 1979 ). In 1878, Hermann Berthold defined 798 points as being equal to 30 cm, or 2660 points equalling 1 meter: that gives around 0.376 mm to the point. A more precise number, 0.376 065  mm , sometimes is given; this is used by TeX as the dd unit. This has become the standard in Germany and Central and Eastern Europe. This size is still mentioned in

1118-600: The royal foot , a legal length measure in France: the Didot point is exactly 1 ⁄ 864 of a French foot or 1 ⁄ 72 of a French inch, that is (by 1799) 15 625 ⁄ 41 559  mm or about 0.375 972  mm . Accordingly, one Didot point is exactly two Truchet points. However, 12 Fournier points turned out to be 11 Didot points, giving a Fournier point of about 0.345 mm ; later sources state it as being 0.348 75  mm . To avoid confusion between

1161-472: The trivial divisors of n . {\displaystyle n.} A divisor of n {\displaystyle n} that is not a trivial divisor is known as a non-trivial divisor (or strict divisor ). A nonzero integer with at least one non-trivial divisor is known as a composite number , while the units −1 and 1 and prime numbers have no non-trivial divisors. There are divisibility rules that allow one to recognize certain divisors of

1204-635: The German, Dutch, French, Polish and all other manuals elsewhere on the European continent for the composition caster and the super-caster are different in quite some details. The Monotype wedges used at the European continent are marked with an extra E behind the set-size: for instance: 5-12E, 1331-15E etc. When working with the E-wedges in the larger sizes the differences will increase even more. The desktop publishing point (DTP point) or PostScript point

1247-472: The eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However, the number of positive divisors is not a totally multiplicative function: if the two numbers m {\displaystyle m} and n {\displaystyle n} share a common divisor, then it might not be true that d ( m n ) = d ( m ) × d ( n ) . {\displaystyle d(mn)=d(m)\times d(n).} The sum of

1290-889: The exact size was not standardized, and various type foundries had been using their own. During and after the American Revolutionary War , Benjamin Franklin was sent as commissioner (Ambassador) for the United States to France from December 1776 to 1785. While living there he had close contact with the Fournier family, including the father and Pierre Simon Fournier . Franklin wanted to teach his grandson Benjamin Franklin Bache about printing and typefounding, and arranged for him to be trained by Francois Ambroise Didot. Franklin then imported French typefounding equipment to Philadelphia to help Bache set up

1333-454: The font's em square, there is not necessarily any size relationship between the two, so the point size does not necessarily correspond to any measurement of the size of the letters on the printed page. The point was first established by the Milanese typographer , Francesco Torniella da Novara ( c.  1490 – 1589) in his 1517 alphabet, L'Alfabeto . The construction of the alphabet

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1376-583: The inch by Donald Knuth for the default unit of his TeX computer typesetting system and is thus sometimes known as the TeX point , which is 0. 351 459 80  mm. Although the English Monotype manuals used 1 pica = .1660 inch, the manuals used on the European continent use another definition: there 1 pica = .1667 inch, the Old English pica. As a consequence all the tables of measurements in

1419-492: The largest of the 23 foundries that would merge in 1892 to form the American Type Founders Co. The official definition of one pica is 0.166044 inches (4.2175 mm), and one point is 0.013837 inches (0.3515 mm). That means 6 picas or 72 points constitute 0.996 24 standard inches. A less precise definition is one pica equals 0.166 inches (4.2 mm), and one point 0.01383 inches (0.351 mm). It

1462-553: The majority of foundries had been using picas less than one sixth of an inch. So in 1886, after some examination of various picas, the Type Founders Association of the United States approved the pica of the L. Johnson & Co. foundry of Philadelphia (the " Johnson pica ") as the most established. The Johnson foundry was influential, being America's first and oldest foundry; established as Binny & Ronaldson in 1796, it would go through several names before being

1505-458: The new and the old sizes, Didot also rejected the traditional names, thus parisienne became corps 5, nonpareille became corps 6, and so on. The Didot system prevailed because the French government demanded printing in Didot measurements. Approximations were subsequently employed, largely owing to the Didot point's unwieldy conversion to metric units (the divisor of its conversion ratio has

1548-686: The positive divisors of n {\displaystyle n} is another multiplicative function σ ( n ) {\displaystyle \sigma (n)} (for example, σ ( 42 ) = 96 = 3 × 4 × 8 = σ ( 2 ) × σ ( 3 ) × σ ( 7 ) = 1 + 2 + 3 + 6 + 7 + 14 + 21 + 42 {\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42} ). Both of these functions are examples of divisor functions . If

1591-543: The print resolution for the LaserWriter printer. In 1996, it was adopted by W3C for Cascading Stylesheets (CSS) where it was later related at a fixed 3:4 ratio to the pixel due to a general (but wrong) assumption of 96 pixel-per-inch screens. Since the advent of high-density "Retina" screens with a much higher resolution than the original 72 dots per inch, Apple's programming environment Xcode sizes GUI elements in points that are scaled automatically to

1634-401: The set N {\displaystyle \mathbb {N} } of non-negative integers into a partially ordered set that is a complete distributive lattice . The largest element of this lattice is 0 and the smallest is 1. The meet operation ∧ is given by the greatest common divisor and the join operation ∨ by the least common multiple . This lattice is isomorphic to the dual of

1677-643: The technical regulations of the Eurasian Economic Union . pdfTEX, but not plain TeX or LaTeX, also supports a new Didot point ( nd ) at 3 ⁄ 8  mm or 0.375 mm and refers to a not further specified 1978 redefinition for it. The French National Print Office adopted a point of 2 ⁄ 5  mm or 0.400 mm in about 1810 and continues to use this measurement today (though "recalibrated" to 0.398 77  mm ). Japanese and German standardization bodies instead opted for

1720-522: The term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned. 1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even , and integers not divisible by 2 are called odd . 1, −1, n {\displaystyle n} and − n {\displaystyle -n} are known as

1763-683: The type of points used. Desktop publishing software and word processors intended for office and personal use often have a list of suggested font sizes in their user interface, but they are not named and usually an arbitrary value can be entered manually. Microsoft Word, for instance, suggests every even size between 8 and 28 points and, additionally, 9, 11, 36, 48 and 72 points, i.e. the larger sizes equal 3, 4 and 6 picas. While most software nowadays defaults to DTP points, many allow specifying font size in other units of measure (e.g., inches, millimeters, pixels), especially code-based systems such as TeX and CSS. Body height (typography) In typography ,

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1806-452: The usual convention that 1 foot equals 12 inches, 1 inch ( pouce ) was divided into 12 lines ( lignes ) and 1 line was further divided into 6 typographic points ( points typographiques ). One Fournier point is about 0.0135 English inches. Fournier printed a reference scale of 144 points over two inches; however, it was too rough to accurately measure a single point. The Fournier point did not achieve lasting popularity despite being revived by

1849-485: Was also noticed that 83 picas is nearly equal to 35 cm, so the Type Founders Association also suggested using a 35 cm metal rod for measurements, but this was not accepted by every foundry. This has become known as the American point system . The British foundries accepted this in 1898. In modern times this size of the point has been approximated as exactly 1 ⁄ 72.27 ( 0.013 837 000 138 37 ) of

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