Vienažodžių matematikos terminų sinonimija matematikos vadovėliuose (1910–1940)

146 Diana Šilobritaitė | Synonymy of Synonymy of single-word mathematical terms in mathematics textbooks (1910–1940) Diana Šilobraitė Institute of Lithuanian Language 1. INTRODUCTION The first mathematical concepts arose in the time of the primitive community from the needs of practical human activity. The term mathematics is believed to have come into use in Greece in the 6th century BC. Kr. Then mathematics became an independent science. Lithuanian mathematicians and linguists began to organize and standardize Lithuanian terms only from the end of the 19th century, although since the end of the 15th century Lithuanian parish schools taught arithmetic, and the Reformed school founded by A. Kulvietis in Vilnius in 1539 taught arithmetic and geometry among the seven liberal arts. The first mathematical works were published in Lithuania in Latin, later in Polish, e.g. : Jono Rudaminos Dusetiškio Garsiosmatematikosteoremosirproblemos (Illustriora theoremataetproblematamathematicaexopticisgeometriaastronomiaspheara elementaricomputoecclesiastico 1633), Osvaldo Krygerio Praktinėaritmetika (Arithmetica practicainusumstudiosaejuventutisAcademiaeetuniversitatis VilnensisSocietatisJesu1635), Mykolo Kazimiero Belkovskio Teorecentrika, arbamatematinissamprotavimasapietaškusircentrus (Theorecentricasive mathematicaedepunctisetcentrisconsiderationes 1644) ir kt. (LTE VII 1981: 308–312). In the middle of the 19th century, the first Lithuanian school textbooks of mathematics were tried to be published. They were prepared by J. Stanevičius (about 1830) and M. P. Marcinskis (about 1860), but they were not published. 1.1. Management of mathematical terminology: textbooks and dictionaries The first Lithuanian textbooks of arithmetic were published at the end of the 19th century: P. Matulionis and J. Spudulis (composed by J. Gailutis) zaduocnis, it is ira Rankius zaduocziu Aritmetikos or Rakundos mokslo (1885), P. Nėris (P. Vileišis) Keturi svarbiausi darbai Aritmetikos (1886), S. Gimžauskas Pradžiamokslis single-word mathematical terms... Terminology | 2007 | 14 147 Lietuvisko spaudraszczo ir rankraszczo (1888), S. Skačkausko Aritmetika (1897). The mathematical terminology of these textbooks was discussed by the researcher of the history of arithmetic terminology P. Rumšas (Rumšas 1978: 52–60). At the beginning of the 20th century, Lithuanian mathematics textbooks and exercises increased – arithmetic began to be taught in the mother tongue. There is a noticeable standardization of terms: "all arithmetic operations are called the same, the numbers with which the operations are performed, the results of those operations, the members of fractions, etc." (Robinson 1980: 73). Several editions of the collection of arithmetic problems and examples by P. Bendorias and P. Daugirda (1906–1918), P. Mašioto arithmetic problem (1906, 2 d.), S. Thomo’s short arithmetic science translated by A. Vėgėlė (1916) appeared. The articles mentioned are: [A. Dambrausko] A. Jakšto A few words about the mathematical criterion of poetic (Draugija 1907 1: 28–31), A. Bernšteinas (translated by B. Lėsys) Some secrets of numbers (Viltis 1908 78: 4; 1908 81: 4), P. Mašiotas From the history of numeracy (Draugija 1908 9: 396–400) and [A. Dambrausko] A. Jakšto review Gradual learning of arithmetic or rokunda (Draugija 1909 27: 272). The first dictionaries of mathematical terms were published in the middle of the period under review – M. Šikšnis’s Dictionary of Arithmetic and Algebra Terms (Vilnius, 1919) and Z. Žemaitis’s Collection of Geometry and Trigonometry Terms (Kaunas, 1920). It was one of the first printed dictionaries of Lithuanian terms. They laid the foundations of Lithuanian school mathematics terminology. The article examines one-word synonyms of mathematical terms, collected from the first half of the 20th century (until 1940) secondary school mathematics textbooks. The material was collected from 24 textbooks – V. Budrevičius’ Theory of Algebra (1919), A. Busila’s Arithmetic (1921), A. Busila’s Young Matematician (1921), J. Damijonaitis’ Arithmetic textbook (1910 (1911), 2 d.), A. Jakšta’s Planar trigonometry (1919), J. Mašiotas’ Mathematics textbook (Geometry, 1937–1938, 3 d.), J. Mašiotas’ Mathematics textbook (Arithmetic and Algebra, 1939–1940, 3 d.), K. Klimavičius’ Geometry (1938–1940, 3 d.), Masingo (translated by A. Smetona) Arithmetic Theory (1922), M. Šikšnis’ Geometry (1923), M. Šikšnis’ Elementary Algebra (1923), M. Šikšnis’ Geometry (1938–1939, 3 d.), M. Šikšnis’ Arithmetic and Algebra (1939, 3 d.). The textbooks of the analyzed period were divided into two categories: theoretical (dedicated only to theory) and mixed (presenting both theory and practice). They were also issued as assignments, but the terms were not chosen from them – they are almost identical to those used in textbooks. 148 Diana Šilobritaitė | Synonymy of 2. SINGLE-WORD SYNONYMS OF MATHEMATICAL TERMS In mathematics textbooks from the 1910s to the 1940s, most of the single-word synonyms are used by different authors. Synonyms used by the same author are fewer. There are two types of synonym strings in terms of origin – multi-origin (synonymy of Lithuanian and foreign-origin terms) and mono-origin (synonymy of purely Lithuanian terms). 2.1. One-word synonyms of mathematical terms used by the same author In the textbook of the same author, only one line of synonyms of Lithuanian terms was found: apibrėžimas (nusakymas) MAT 1922 11. The dictionaries of Z. Žemaitis and M. Šikšnis contain only the definition of the term, which has been prepared by J. Jablonskis based on the verb to define “to provide a definition”. (see Rumšas 1975: 8). In the textbooks of the same authors, more frequent lines of synonyms of different origins are: binary – binomial ŠAA I 1939 110; contiguous – parallelepiped BJM 1921 79; in-beam – hypotenuse BJM 1921 17, KG 1939 19, KG 1940 31, MG I 1937 52, MG II 1938 4, MG III 1938 3, ŠG 1923 25, ŠG I 1938 36, ŠG II 1938 55, ŠG III 1939 4; cube – hexagonal MAT 1922 46. In mathematical terminology, the Lithuanian terms gretasienis, which has been proposed by Z. Žemaitis, and įžambinė have become established and are used. MTŽ 1994 binary and binomial are still presented as synonyms: binomial see binary – binary, binomial: differential, linear, but in some generic names they are distinguished: binomial cube, binomial square and binomial spread, binomial formula. A rectangular adjacent, whose sides are squares, is called a non-Lithuanian term cube. The terms parallelepiped (Russian: параллелепипед, English: parallelepiped) and hypotenuse (Russian: гипотенуза, English: hypotenuse) have not been established in Lithuanian mathematical terminology and are not used. 2.2. Single-word synonyms of mathematical terms used by different authors The Lithuanian synonyms of terms used by different authors are lexical and constructive. A synonym is a word that has the same meaning, but is used to denote the same thing or person (Pikčilingis 1969: 4). single-word mathematical terms... Terminology | 2007 | 14 149 Lexical synonyms are the most common in the mathematics textbooks examined. 2.2.1. The following are some of the most commonly used synonyms of terms in different languages: diagonal BJM 1921 3, SG I 1938 34 – diagonal BJM 1921 3, SG 1923 24; cut BA 1921 4, KG 1938 19, MG I 1937 26, SG 1923 66, SG I 1938 16, SG III 1939 24 – sector SG 1923 66 / sector SG III 1939 24; mat BJM 1921 9, JPT 1919 7, KG 1938 28, KG 1939 5, KG 1940 70, MAA I 1940 142, MG I 1937 36, SG I 1938 21 – transporter SG I 1938 21; cut KG 1938 21, MG I 1937 26, SG 1923 66, SG I 1938 16, SG III 1939 25 – segment SG 1923 66, SG III 1939 25; semi-circular KG 1938 46, KG 1940 87, MAA III 1940 66, SG 1923 15, SG I 1938 40 – bi-sector KG 1938 46, SG 1923 15; semi-circular KG 1938 46, SG 1923 26 – median JPT 1919 104, KG 1938 46, SG 1923 26; flying BA 1921 3, BJM 1921 30, JPT 1919 46, KG 1938 13, KG 1940 104, MG I 1937 19, MG II 1938 124, MG III 1938 75, SG 1923 10, SG I 1938 10 – circle KG 1938 51, SG 1923 10; static BJM 1921 17, KG 1938 44, KG 1939 12, KG 1940 17, MAA II 1939 117, MG I 1937 52, MG II 1938 4, MG III 1938 3, SG 1923 25 – catheter BJM 1921 17, KG 1938 44, SG III 1923 25; string BJM 1921 52, KG 1938 20, KG 1939 32, KG 1940 45, MG I 1937 26, MG II 1938 48, MG III 1938 55, ŠG 1923 66, ŠG I 1938 16, ŠG II 1938 18, ŠG III 1939 5 – chord BJM 1921 52, JPT 1919 61, ŠG 1923 66. In Z. Žemaičio Geometry and trigonometry terminology all these terms are also presented as synonyms. In modern mathematical terminology, the terms bisectrix (Russian: биссектриса), diagonal (Russian: диагональ), transporter (Russian: транспортир), circle (Russian: циркуль), cathet (Russian: катет), chorda (Russian: хорда) are not used. The semi-circular, diagonal, matilanki, flying, static, string (MTZ 1994) were introduced. Synonyms are considered to be notching – sector, notching – segment, but the main ones are Lithuanian terms notching, notching. In mathematics, the median is the mean value of a random variable (in probability theory and mathematical statistics) and the semi-median is the mean value of a random variable (in probability theory and mathematical statistics). Z. Žemaitis does not use terms of non-Lithuanian origin: kūgis BJM 1921 89, KG 1938 5, MG I 1937 8 – kūnas BJM 150 Diana Šilobritaitė | Synonyms of one-word mathematical terms... 1921 MAA III 1940 66, MG I 1937 46, MG II 1938 5, MG III 1938 53, ŠAA II 1939 24, ŠG 1923 15, ŠG I 1938 45, ŠG II 1938 10, ŠG III 1939 3 – perpendicular ŠG 1923 15. They are not used in modern mathematical terminology. In Z. Žemaičio Geometry and Trigonometry Terms Collection, the fourth BJM 1921 19, MAT 1922 44 – square BA 1921 7, BJM 1921 19, KG 1938 84, KG 1939 19, KG 1940 7, MAA I 1940 22, MAA III 1940 4, MAT 1922 44, MG I 1937 109, MG II 1938 3, ŠG 1923 62, ŠAA I 1939 44, ŠAA II 1939 3, ŠAA III 1939 11, ŠEA 1923 14 are not considered synonyms. The fourth and fifth floors are decorated with a quadrangle-shaped portico. M. Šikšnys does not use the term cifra in the Dictionary of Arithmetic and Algebraic Terms, but in later years the digit is used in textbooks: BA 1921 5, BAT 1919 6, BJM 1921 9, KG 1938 22, MAA I 1940 11, MAA III 1940 46, MAT 1922 9, ŠG 1923 12, ŠAA I 1939 6, ŠAA II 1939 101, ŠAA III 1939 13, ŠG I 1938 81, ŠG III 1939 6 – cifra BAT 1919 5. In M. Šikšnio glossary, the terms used in textbooks are considered synonyms: BAT 1919 31, MAT 1922 11 – sum BA 1921 10, BAT 1919 17, BJM 1921 4, DAV 1910 39, JPT 1919 40, KG 1938 14, KG 1940 20, MAA I 1940 41, MAA III 1940 12, MAT 1922 11, MG I 1937 20, MG II 1938 6, MG III 1938 23, ŠAA I 1939 10, ŠAA II 1939 5, ŠAA III 1939 4, ŠEA 1923 10, ŠG 1923 11, ŠG I 1938 11, ŠG II 1938 33, ŠG III 1939 4. In fact, M. Šikšnys still presents the compound, but in his textbooks he uses only the term sum, which has become established in mathematics. The quaternary row of synonyms usually consists of one international and three Lithuanian terms: cross section JPT 1919 28 – cross section BA 1921 50, BJM 1921 44 – diameter BA 1921 50, KG 1938 20, KG 1940 38, MAA II 1939 17, MG I 1937 26, MG III 1938 53, ŠG 1923 66, ŠG I 1938 16, ŠG II 1938 17, ŠG III 1939 5 – diameter BJM 1921 89, JPT 1919 18, KG 1938 20, ŠG 1923 66. Individual authors used crossword, crossword did not succeed. The diameter of the tube is not measured. The fifth row consists of four Lithuanian and one non-Lithuanian term: mastas DAV 1910 20 – scale KG 1940 39, MAA I 1940 22, MG II 1938 119, MG III 1938 4, ŠAA 1939 48, ŠG II 1938 68 – scale KG 1940 39 – scale BA 1921 46, BJM 1921 35 – measure BJM 1921 39. As can be seen, the most common is the term scale, which was used in later 89; perpendicular BJM 1921 3, KG 1938 32, KG 1939 12, KG 1940 31, MAA II 1939 34, 2007 | 14,151 in textbooks (1938, 1940). However, there is still the term dydrodis, which was later abandoned. Z. Žemaitis and J. Jablonskis suggested the scale as a name for the scale (see Klimavičius 2005: 143–146). One case was found where the synonym string consists of hybrid mono-word terms, e.g.: ruler BJM 1921 12, KG 1938 7, KG 1940 70, MAA II 1939 60, MG I 1937 11, MG III 1938 90, ŠAA II 1939 32, ŠG 1923 7, ŠG I 1938 7, ŠG III 1939 67 – ruler JPT 1919 46. As you can see, the term ruler is used by only one author. This term has not survived to our days, and in mathematical terminology the term ruler is used. 2.2.2. Synonyms of Mathematical Terms There are many cases where one of the synonyms is used by most of the authors of mathematics textbooks and the other by only one: circle BJM 1921 51, KG 1938 17, KG 1939 30, KG 1940 7, MAA II 1939 149, MG I 1937 11, MG II 1938 3, ŠG 1923 43, ŠG I 1938 15, ŠG II 1938 17, ŠG III 1939 3 – rim ŠG 1923 43 (in the Z. Žemaičio collection of terms of geometry and trigonometry, synonyms are circle and edge); section BA 1921 4, BJM 1921 23, KG 1938 10, KG 1939 5, KG 1940 5, MAA I 1940 11, MAA II 1939 32, MG I 1937 17, MG III 1938 21, ŠAA I 1939 8, ŠAA II 1939 26, ŠEA 1923 27, ŠG I 1938 10 – cut-off JPT 1919 28 (in the Z. Žemaičio collection only a section); product MAA I 1940 15, MAA II 1939 9, MAA III 1940 11, ŠAA I 1939 156, ŠAA III 1939 3 – product of MAA I 1940 15 (Z. Žemaitis only product); ball BJM 1921 76, KG 1938 5, MAA II 1939 24, MG I 1937 9 – ball KG 1938 5 (in Z. Žemaitis’ terminology – ball); remnant BA 1921 9, BAT 1919 37, JPT 1919 30, DAV 1910 27, KG 1938 12, MAA I 1940 35, MAA III 1940 34, ŠAA I 1939 6, ŠAA II 1939 101, ŠAA III 1939 15 – editions MAT 1922 30 (M. Šikšnys and Z. Žemaitis give both terms in their dictionaries); equation BAT 1919 87, MAA II 1939 6, MG III 1938 6, ŠAA II 1939 50, ŠAA III 1939 11, ŠG III 1939 26 – equation JPT 1919 9, MAT 1922 128 (in the Z. Žemaitis collection these terms are not considered synonyms: equation (Russian: уравнение) and equation (равенство (формула)); condition BJM 1921 38, JPT 1919 9, KG 1938 64, MAT 1922 110 – contract MAT 1922 110; the product of BA 1921 10, BAT 1919 17, BJM 1921 45, KG 1939 12, KG 1940 7, MAA I 1940 48, MAA II 1939 15, MAA III 1940 3, MAT 1922 21, MG I 1937 41, MG II 1938 11, MG III 1938 14, ŠG 1923 11, ŠAA I 1939 7, ŠAA II 1939 9, ŠAA III 1939 3 – the product of JPT 1919 53 (in the M. Šikšnio glossary – the product); meter BA 1921 5, BAT 152 Diana Šilobritaitė | Synonymy of single-word mathematical 1939 107 – numerator DAV 1910 25 (in M. Šikšnio dictionary of terms – numerator); circle BA 1921 3, BJM 1921 51, JPT 1919 13, KG 1938 7, KG 1939 37, MAA I 1940 22, MG I 1937 11, MG III 1938 53, ŠG 1923 66, ŠG I 1938 5, ŠG III 1939 22 – circle ŠG 1923 66 (in Z. Žemaičio terminology – circle). The first dictionaries of mathematical terms are used by many mathematical textbook authors, but some authors have later added their own terms. Z. Žemaitis, speaking about the term lygtis, pointed out that it was not present in earlier Lithuanian writings. Therefore, it is not surprising that the term "comparison" was used to describe this concept, especially in earlier textbooks. Jablonski suggested that the term equation should be used not in the singular but in the plural. In his opinion, “here we have to deal with the necessary interaction, equality and balance of the two parts” (Žemaitis 1966: 198). However, mathematicians convinced Jablonski that “it is important to have the concept expressed in singular, so that it is clear that we are dealing not with several, but with one equation” (Žemaitis 1966: 198). The member of the string of lexical synonyms of the trinar is the denominator used most often in the mathematics textbooks of the period under analysis: denominator BAT 1919 17, JPT 1919 22, KG 1940 71, MAA I 1940 34, ŠAA I 1939 8, ŠAA II 1939 63 – marker DAV 1910 25 – marker BA 1921 5. It is also included in the M. Šikšnis's Dictionary of Arithmetic and Algebra Terms. The term marker is used in a textbook (J. Damijonaitis) published before the publication of M. Šikšnis’s dictionary (in the textbook of P. Matulionis and J. Spudulis [composed by J. Gailutis] it is the marker used in the textbook of Aritmetika or Rakundos mokslo (1885)). The denominator is already used in the 1919 textbook (it is also presented in M. Šikšnis's Dictionary of Arithmetic and Algebraic Terms), so it is not clear why the term marker is used in the 1921 textbook to mark this concept. According to J. Vaskelienė, verb synonyms are those that have the same root and are derivatives of the same verb category with a close meaning (Vaskelienė 2000: 75). Often the terms of the synonym lines differ by the prefixes of the division, e.g.: dalik lis BA 1921 13, BAT 1919 17, MAA I 1940 72, MAA II 1939 5, MAT 1922 29, ŠAA I 1939 6, ŠAA II 1939 7, ŠAA III 1939 6, ŠEA 1923 38 – divisor DAV 1910 27; multiplication BA 1921 21, BAT 1919 16, JPT 1919 86, terms... 1919 17, JPT 1919 22, KG 1940 71, MAA I 1940 34, ŠAA I 1939 8, ŠAA II Terminology | 2007 | 14 153 MAA I 1940 41, MAA III 1940 13, MAT 1922 21, ŠAA I 1939 31, ŠAA II 1939 97, ŠAA III 1939 9, ŠEA 1923 7, ŠG 1923 11, ŠG I 1938 12 – multiplication BA 1921 29, DAV 1910 4, DAV 1911 8, JPT 1919 74, MAT 1922 24; law BJM 1921 33, KG 1938 8 – taught MAT 1922 47. The divisor has not been adopted in mathematical terminology because it contradicts the laws of word formation: with the suffix -tojas usually form names of actors and much less frequently names of possessors of verb properties. This term was used in the first mathematics textbooks – P. Matulionis and J. Spudulis (composed by J. Gailutis) (1885), P. Nėris (P. Vileišis) (1886). After the publication of M. Šikš nio’s Dictionary of Arithmetic and Algebraic Terms, the divisor was consistently used. The terms multiplication and multiplication (which were used until the 20th century) are still used in modern terminology. The term multiplication, according to Z. Žemaitis, must be used to “denote the execution of specific actions” (1966) 1966. To this group can be assigned synonyms lines, one member of which is a conjugated participant and the other a noun of the prepositional action: divisible BA 1921 52, BAT 1919 17 – subunit BA 1921 29, BAT 1919 52, DAV 1910 27, JPT 1919 53, KG 1938 14, MAA I 1940 72, MAA II 1939 14, MAA III 1940 15, MAT 1922 29, ŠAA I 1939 33, ŠAA II 1939 7, ŠAA III 1939 9, ŠEA 1923 7, ŠG I 1938 12, ŠG III 1939 6 (in M. Šikšnis’s glossary of arithmetic and algebra terms these terms are also given as synonyms); touching BJM 1921 53, JPT 1919 29, ŠG 1923 72, ŠG II 1938 17 – casting KG 1939 35, KG 1940 52, MG II 1938 51, ŠG II 1938 17. There has been considerable controversy over the term. Z. Žemaitis noted that the term tangent can cause ambiguity (but he presented it in the Geometry and Trigonometry Terms Book): "is it a line that touches something, or which something touches?" J. Jablonskis proposed the term litėja "emphasizing a certain activity of it" (Žemaitis 1966: 198). In modern mathematical terminology, a line that has only one point in common with a curve is called tangent (MTZ 1994). Another row of synonyms consists of the noun form of the adjective and the participle: the slant BJM 1921 53, MG I 1937 42, ŠG I 1938 28, ŠG III 1939 40 – the slant KG 1938 32, ŠG 1923 15. The term is still used today in mathematical terminology to refer to a straight or plane oblique line (MTŽ 1994). The rectangular KG 1938 82, ŠG 1923 60 – rectangle BA 1921 7, BJM 1921 19, KG 1939 5, KG 1940 31, MAA I 1940 22, MAA III 1940 153, MG I 1937 105, MG II 1938 3, ŠG are also considered to be operative synonyms. 154 Diana Šilobritaitė | Synonymy of single-word mathematical terms... 1938, pp. 35. In triangle. In mathematics textbooks, threeand four-letter synonyms are found: crossing BJM 1921 52, KG 1940 9, ŠG 1923 66, ŠG II 1938 17 – crossing MAT 1922 44 – crossing KG 1939 34, KG 1940 52, MG II 1938 50, ŠG I 1938 16, ŠG II 1938 17; composition BA 1921 72, BAT 1919 12, MAA I 1940 41, MAA II 1939 28, MAA III 1940 12, MAT 1922 11, ŠAA I 1939 20, ŠAA II 1939 93, ŠEA 1923 7, ŠG I 1938 11 – addition MAT 1922 12 – addition DAV 1910 3, DAV 1911 3; division BA 1921 13, KG 1938 14, MAA I 1940 41, MAA III 1940 15, MAT 1922 29, ŠAA I 1939 31, ŠAA III 1939 9, ŠEA 1923 7, ŠG I 1938 12, ŠG III 1939 6 – division BA 1921 27, BAT 1919 65, DAV 1910 4, DAV 1911 9, ŠAA I 1939 157 – division JPT 1919 74 – division MAT 1922 64; multiplier BJM 1921 82, JPT 1919 128, MAA I 1940 48, MAA II 1939 5, MAT 1922 21, MG I 1937 21, ŠEA 1923 11, ŠG I 1938 12, ŠG III 1939 23 – multiplicative BA 1921 16, BAT 1919 17, MAA I 1940 48, MAA II 1939 21, MAA III 1940 3, ŠAA I 1939 7, ŠAA II 1939 3, ŠAA III 1939 4 – plural of ŠAA I 1939 31 – plural BA 1921 18, BAT 1919 41, MAA I 1940 48, MAT 1922 21, ŠEA 1923 36. As can be seen, synonyms are used here in textbooks of the same and different authors. For a quantity that is multiplied by another quantity, Jablonski proposed the terms multiple (for the first number) and multiplier (for the second number). P. Rumšas notes that “plural and dalmuo are new forms of the same prefix, so it is not clear why plural had to mean the multiplier, and dalmuo, on the contrary, not the divisor, but the resulting number” (Rumšas 1975: 7). Not all of the terms in the above synonymous rows have been adopted and are used in mathematics. For example, kertamoji – kertė – kirtėja: kertamoji – “a machine for harvesting grain”, kertė – “the corner of a room or object; small plot, outskirts”, kirtėja – a noun naming a person. In mathematics, the term intersecting line (MTZ 1994) is used, but the term intersecting is used to refer to a line that intersects a curve at at least two points. The term is also used in mathematical terms to describe the ability of a mathematical system to "resolve" a problem (e.g., to resolve a problem). Division and division have different meanings. The term "multiplication" proposed by J. Jablonski was a novelty – it has only one meaning (mathematical): multiplication as "multiplication action" (J. Jablonski): multiplication table, multiplication alphabet (Z. Žemaitis). Multiplicity Widgets modern mathematics, the term is used to refer to the rectangular shape of a