Linear A incised on tablets found in Akrotiri, Santorini.

Linear A is one of two scripts used in ancient Crete before Mycenaean Greek Linear B; Cretan hieroglypic is the second script. In Minoan times, before the Mycenaean Greek dominion, Linear A was the official script for the palaces and religious activities, and hieroglyphs were mainly used on seals. These three scripts were discovered and named by Arthur Evans. In 1952, Michael Ventris discovered that Linear B was being used to write the early form of Greek known as Mycenaean. He and others used this information to achieve a significant and now well-accepted decipherment of Linear B, although many points remain to be clarified. By contrast, Linear A has not been deciphered since the language of Linear A has not been discovered.

Though the two scripts – Linear A and B – share some of the same symbols, using the syllables associated with Linear B in Linear A writings produces words that are unrelated to any known language. This language has been dubbed Minoan and corresponds to a period in Cretan history prior to a series of invasions by Mycenaean Greeks around 1450 BC.

Linear A seems to have been used as a complete syllabary around 1900–1800 BC, although several signs appear earlier as mason marks. It is possible that the Trojan Linear A scripts that were discovered by Heinrich Schliemann and one inscription from central Crete, as well as a few similar potters' marks from Lahun, Egypt (12th dynasty), come from an earlier period, ca. 2100–1900 BC, which coincides with the construction of the first palaces.

Theories of decipherment[link]

Linear A incised on a vase, also found in Akrotiri.

As the Minoan language is lost to the modern day, it is hard to be certain whether or not a given decipherment is correct. The simplest approach to decipherment may be to presume that the values of Linear A match more or less the values given to the fully translated Linear B script, used for Mycenean Greek.^[2] This point of view has been of great interest to archaeologists. The lack of a decipherment means there is no definite evidence for this view, though there is support based on onomatopoeia (for example: AB 23 'mu' from Hieroglyphic 12, a bull-head; AB 80 'ma' from H cat-head; AB 67 'ki' from H 57 a sistrum (a type of rattle); AB 60 'ra' from H 18 dog-head; AB 50 'pu' from H 58 harp). In addition, complex words of three or more syllables appear in both Linear A and B (therefore, 12 signs have the same values in both syllabaries: DA, I, JA, KI, PA, PI, RO, RI, SE, SU, TA, O).

One of the very few understood words so far, the summarizing term, KU-RO, most likely meaning 'total' or something similar to it, could be of either Indo-European *kwol- (o-grade form of *kwel-^[3], cognate to English "whole"), or Semitic (*kull- 'whole') origin, or a language isolate, unrelated to either. This ambiguity represents the current state of understanding of the language of Linear A: the known elements are too scarce to build a safe hypothesis about supposed affiliations with known languages.

Greek[link]

Vladimir I. Georgiev published his Le déchiffrement des inscriptions crétoises en linéaire A ("The decipherment of Cretan inscriptions in Linear A") in 1957 stating that Linear A contains Greek linguistic elements.^[4] In 1963, he published an article, "Les deux langues des inscriptions crétoises en linéaire A" ("The two languages of Cretan inscriptions in Linear A"), suggesting that the language of the Hagia Triada tablets was Greek but that the rest of the Linear A corpus was in Hittite-Luwian.^[4]

Luwian[link]

Since the 1960s, a theory based on Linear B phonetic values suggests that Linear A language could be an Anatolian language, close to Luwian.^[5] In 1997, Gareth Alun Owens published a collection of essays entitled Kritika Daidalika, which support the view that Linear A might represent an archaic relative of Luwian. Owens based this assertion on the perceived Indo-European but non-Greek roots of a small number of words he was able to read by using the known Linear B or Cypriot sound values of certain Linear A signs. He does not claim a systematic decipherment of Linear A, and remarks in the book that he intended his Luwian hypothesis to provoke discussion rather than to settle the issue.

The theory for the Luwian origin of Minoan, however, failed to gain universal support for the following reasons:

• There is no remarkable resemblance between Minoan and Hitto-Luwian morphology.
• None of the existing theories of the origin of Hitto-Luwian peoples and their migration to Anatolia (either from the Balkans or from the Caucasus) is related to Crete.
• There was a lack of direct contacts between Hitto-Luwians and Minoan Crete; the latter was never mentioned in Hitto-Luwian inscriptions. Small states located along the western coast of ancient Asia Minor were natural barriers between Hitto-Luwians and Minoan Crete.
• Obvious anthropological differences between Hitto-Luwians and the Minoans may be considered as another indirect testimony against this hypothesis.

Phoenician[link]

In 2001, the journal Ugarit-Forschungen, Band 32 published the article "The First Inscription in Punic — Vowel Differences in Linear A and B" by Jan Best, claiming to demonstrate how and why Linear A notates an archaic form of Phoenician.^[6] This was a continuation of attempts by Cyrus Gordon in finding connections between Minoan and West Semitic languages. His methodology drew widespread criticism. While one or two terms may apparently be of Semitic origin (such as KU-RO, see below), there is yet not enough evidence to secure a connection between the language of Linear A and Semitic idioms. Contrary to most other scripts used for Semitic languages, Linear A presents many written vowels.^{[citation needed]}

Indo-Iranian[link]

Another recent interpretation, based on the frequencies of the syllabic signs and on complete palaeographic comparative studies, suggests that the Minoan Linear A language belongs to the Indo-Iranian family of Indo-European languages.^[7] Studies by Hubert La Marle include a presentation of the morphology of the language, avoid the complete identification of phonetic values between Linear A and B, and also avoid comparing Linear A with Cretan Hieroglyphs.^[7] La Marle uses the frequency counts to identify the type of syllables written in Linear A, and takes into account the problem of loanwords in the vocabulary.^[7] However, the La Marle interpretation of Linear A has been rejected by John Younger of Kansas University showing that La Marle has invented erroneous and arbitrary new transcriptions based on resemblances with many different script systems at will (as Phoenician, Hieroglyphic Egyptian, Hieroglyphic Hittite, Ethiopian, Cypro-Minoan, etc.), ignoring established evidence and internal analysis, while for some words he proposes religious meanings inventing names of gods and rites.^[8][9]

Tyrrhenian[link]

G. M. Facchetti has attempted to link Linear A to the Tyrrhenian language family comprising Etruscan, Rhaetic, and Lemnian. This family is reasoned to be a pre-Indo-European Mediterranean substratum of the 2nd millennium BC, sometimes referred to as Pre-Greek, and this is supported by confirmation from ancient Greek authors like Herodotus in Histories that the Etruscans were from Lydia (SW Turkey).^{[citation needed]} G. M. Facchetti proposed some possible similarities between the Etruscan language and ancient Lemnian, and other Aegean languages like Minoan.^[10] Michael Ventris who, along with John Chadwick, successfully deciphered Linear B, also believed in a link between Minoan and Etruscan.^[11]

Nature of the texts[link]

A stone ladle from Troullos (TL Za 1) is a likely example of a votive text that is read according to the hypothesis that Linear A values are equal to Linear B values:

a-ta-i-*301-wa-ja o-su-qa-re ja-sa-sa-ra-me u-na-ka-na-si i-pi-na-ma si-ru-te

While the Haghia Triada tablet 13 (HT 13) is an example of an accounting text:

ka-u-de-ta [wine ideogram]. te. re-za 5½ te-ro2 56 te-ki 27½ ku-dzu-ni 18 da-si-*118 19 ?-su-?-si 5 ku-ro 130½

• ka-u-de-ta is followed by an ideogram almost identical to one in Linear B meaning 'wine'. These are followed by a list of six names each followed by a numeral, and then a total (ku-ro) of the preceding numbers — slightly in error, as the correct sum is 131.^[12]

Glossary[link]

This glossary contains terms that are possible meanings according to the rule that Linear A values are the same symbolically and phonetically as Linear B values. The following values remain conjectural because of the paucity of lengthy Linear A texts available:

• A-KU-TU / A-KO-TO: name, Άκτωρ(?).^[13]
• (J)A-DI-KI-TE-TE / JA-DI-KI-TU: Mount Dikte.
• DI-KI-SE / DE-KE-SE-U: name, Dexeus or Derxeus (derived from δέχομαι or δέρκομαι).^[14]
• DU-PU2-RE: ruler, master.
• I-DA: Mount Ida (Crete).
• I-JA-TE: iātēr (Homeric Greek: ιητήρ), a word which occurs in Linear B as well.^[15]
• I-TA-NU / U-TA-NO: place name, probably the Cretan city of Itanos.^[16]
• KA-DA-NA / KA-DA-NO: name, Χαλδάνος(?).^[16]
• KA-RU / KA-RO: name, Κάλλων or Χαίρων or Χάρων(?).^[16]
• KI-RO: missing, debt (possibly Greek: khreos "debt").
• KU-MI-NA: Semitic loanword meaning "cumin"; also appears in Linear B documents.^[17]
• KU-RO: whole, total (vel. sim.) (Cyrus Gordon: < Semitic *kull-; or Greek holon "total"/"whole").
• KU-RU-KU / KU-RU-KA: name, Γλύκη(?).^[18]
• MA-DI: name, Μαίδος(?) from *Maidos to Maidos.^[19]
• MA-RI-TA: ethnic Malitās (Greek: Μαλίτης "Melian").^[14]
• MA-RU / MA-RO: name, Μάρων(?).^[13]
• PA-DE: a theonym (name for a god), appearing on Linear B tablets as well (as pa-de / pa-ze); Cf. Sanskrit pati, lord, Greek Pater, father, master.
• PA-I-TO: place name, Phaistos; the same name is common on Linear B documents.^[15]
• PO-TO-KU-RO: grand(?) total (vel. sim.).
• RO+JA: Royalty, often associated with Rhea.
• SE-TO-I-JA: place name, which occurs in Linear B as well.
• SU-KI-RI-TA: Sugrita, a place name which occurs in Linear B as well; the town survives today as Sybrita.^[15]
• SU-KI-RI-TE-I-JA: possibly "Sugritaian".^[12]

Apart from these, there are a considerable number of proper names and related elements occurring both in Linear A and Linear B namely in the Mycenaean texts from Knossos.

Sites yielding Linear A inscriptions[link]

Timeline of Cretan scripts[link]

The sequence and the geographical spread of Cretan hieroglyphs, Linear A and Linear B, the three overlapping, but distinct writing systems on Bronze Age Crete and the Greek mainland can be summarized as follows:^[20]

See also[link]

• Aegean numbers
• Linear C
• Phaistos Disc

References[link]

Notes[link]

Citations[link]

Sources[link]

• Chadwick, John. The Decipherment of Linear B. Cambridge University Press. ISBN 0-521-39830-4. 
• Nagy, Gregory (1963). "Greek-Like Elements in Linear A". Greek, Roman, and Byzantine Studies (Harvard University Press) (4): 181–211. 
• Olivier, J. P. (1986). "Cretan Writing in the Second Millennium B.C.". World Archaeology 17 (3): 377–389. DOI:10.1080/00438243.1986.9979977. 
• Palaima, Thomas G. (1989). "Cypro-Minoan Scripts: Problems of Historical Context". In Duhoux, Yves; Palaima, Thomas G.; Bennet, John. Problems in Decipherment. Louvain-La-Neuve: Peeters. pp. 121-188. ISBN 90-6831-177-8. http://books.google.com/books?id=TanjAHRuJt4C&pg=PA123&lpg=PA123&dq=enkomi+tablets&source=web&ots=k-Zm8TB2PO&sig=hokXDifykRCR5fRTaEX6YmNE5aA#PPA121,M1. 

Further reading[link]

• Best, Jan G. P. (1972). Some Preliminary Remarks on the Decipherment of Linear A. Hakkert. ISBN 90-256-0625-3. 
• Woodard, Roger D. (1997). Greek Writing from Knossos to Homer. New York: Oxford University Press. ISBN 0-19-510520-6.  (Review.)

External links[link]

Wikimedia Commons has media related to: Linear A

• Linear A Texts in Phonetic Transcription by John Younger (last update: 20 February 2010).
• DAIDALIKA - Scripts and Languages of Minoan and Mycenaean Crete
• Omniglot: Writing Systems & Languages of the World
• Mnamon: Antiche Scritture del Mediterraneo, (Antique writings of the Mediterranean) - a guide to online resources, including a link to GORILA
• GORILA volume 1

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2007)

In mathematics, a linear map or linear function f(x) is a function which satisfies the following two properties:

• Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition.
• Homogeneity of degree 1: f(αx) = αf(x) for all α.

It can be shown that additivity implies the homogeneity in all cases where α is rational; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers. If f is assumed to be continuous as well then this can be extended to show that homogeneity for α any real number, using the fact that rationals form a dense subset of the reals.

In this definition, x is not necessarily a real number, but can in general be a member of any vector space. A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.

The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it is particularly easy to solve by breaking the equation up into smaller pieces, solving each of those pieces, and adding the solutions up.

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations (also called linear maps), and systems of linear equations.

The word linear comes from the Latin word linearis, which means pertaining to or resembling a line. For a description of linear and nonlinear equations, see linear equation. Nonlinear equations and functions are of interest to physicists and mathematicians because they can be used to represent many natural phenomena, including chaos.

Integral linearity[link]

For a device that converts a quantity to another quantity there are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics.

Many times a device's specifications will simply refer to linearity, with no other explanation as to which type of linearity is intended. In cases where a specification is expressed simply as linearity, it is assumed to imply independent linearity.

Independent linearity is probably the most commonly-used linearity definition and is often found in the specifications for DMMs and ADCs, as well as devices like potentiometers. Independent linearity is defined as the maximum deviation of actual performance relative to a straight line, located such that it minimizes the maximum deviation. In that case there are no constraints placed upon the positioning of the straight line and it may be wherever necessary to minimize the deviations between it and the device's actual performance characteristic.

Zero-based linearity forces the lower range value of the straight line to be equal to the actual lower range value of the device's characteristic, but it does allow the line to be rotated to minimize the maximum deviation. In this case, since the positioning of the straight line is constrained by the requirement that the lower range values of the line and the device's characteristic be coincident, the non-linearity based on this definition will generally be larger than for independent linearity.

For terminal linearity, there is no flexibility allowed in the placement of the straight line in order to minimize the deviations. The straight line must be located such that each of its end-points coincides with the device's actual upper and lower range values. This means that the non-linearity measured by this definition will typically be larger than that measured by the independent, or the zero-based linearity definitions. This definition of linearity is often associated with ADCs, DACs and various sensors.

A fourth linearity definition, absolute linearity, is sometimes also encountered. Absolute linearity is a variation of terminal linearity, in that it allows no flexibility in the placement of the straight line, however in this case the gain and offset errors of the actual device are included in the linearity measurement, making this the most difficult measure of a device's performance. For absolute linearity the end points of the straight line are defined by the ideal upper and lower range values for the device, rather than the actual values. The linearity error in this instance is the maximum deviation of the actual device's performance from ideal.

Linear polynomials[link]

In a different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line.

Over the reals, a linear equation is one of the forms:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = m x + b\

where m is often called the slope or gradient; b the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.

Note that this usage of the term linear is not the same as the above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if b = 0. Hence, if b ≠ 0, the function is often called an affine function (see in greater generality affine transformation).

Boolean functions[link]

In Boolean algebra, a linear function is a function Failed to parse (Missing texvc executable; please see math/README to configure.): f

for which there exist Failed to parse (Missing texvc executable; please see math/README to configure.): a_0, a_1, \ldots, a_n \in \{0,1\}
such that 

Failed to parse (Missing texvc executable; please see math/README to configure.): f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \cdots \oplus (a_n \land b_n)

for all Failed to parse (Missing texvc executable; please see math/README to configure.): b_1, \ldots, b_n \in \{0,1\}.

A Boolean function is linear if one of the following holds for the function's truth table:

1. In every row in which the truth value of the function is 'T', there are an odd number of 'T's assigned to the arguments and in every row in which the function is 'F' there is an even number of 'T's assigned to arguments. Specifically, f('F', 'F', ..., 'F') = 'F', and these functions correspond to linear maps over the Boolean vector space.
2. In every row in which the value of the function is 'T', there is an even number of 'T's assigned to the arguments of the function; and in every row in which the truth value of the function is 'F', there are an odd number of 'T's assigned to arguments. In this case, f('F', 'F', ..., 'F') = 'T'.

Another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference.

Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.

Physics[link]

In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation.

Linearity of a differential equation means that if two functions f and g are solutions of the equation, then any linear combination af+bg is too.

Electronics[link]

In electronics, the linear operating region of a device, for example a transistor, is where a dependent variable (such as the transistor collector current) is directly proportional to an independent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, linear regulators, and linear amplifiers in general.

In most scientific and technological, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort even a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value, taking it away from the approximately linear part of the transfer function.

Military tactical formations[link]

In military tactical formations, "linear formations" were adapted from phalanx-like formations of pike protected by handgunners towards shallow formations of handgunners protected by progressively fewer pikes. This kind of formation would get thinner until its extreme in the age of Wellington with the 'Thin Red Line'. It would eventually be replaced by skirmish order at the time of the invention of the breech-loading rifle that allowed soldiers to move and fire independently of the large scale formations and fight in small, mobile units.

Art[link]

Linear is one of the five categories proposed by Swiss art historian Heinrich Wölfflin to distinguish "Classic", or Renaissance art, from the Baroque. According to Wölfflin, painters of the fifteenth and early sixteenth centuries (Leonardo da Vinci, Raphael or Albrecht Dürer) are more linear than "painterly" Baroque painters of the seventeenth century (Peter Paul Rubens, Rembrandt, and Velázquez) because they primarily use outline to create shape.^[1] Linearity in art can also be referenced in digital art. For example, hypertext fiction can be an example of nonlinear narrative, but there are also websites designed to go in a specified, organized manner, following a linear path.

Music[link]

In music the linear aspect is succession, either intervals or melody, as opposed to simultaneity or the vertical aspect.

Measurement[link]

In measurement, the term "linear foot" refers to the number of feet in a straight line of material (such as lumber or fabric) generally without regard to the width. It is sometimes incorrectly referred to as "lineal feet"; however, "lineal" is typically reserved for usage when referring to ancestry or heredity.[1] The words "linear"[2] & "lineal" [3] both descend from the same root meaning, the Latin word for line, which is "linea".

See also[link]

• Linear element
• Linear system
• Linear medium
• Linear programming
• Bilinear
• Multilinear
• Linear motor
• Linear A and Linear B scripts.
• Linear interpolation

References[link]

1. ^ Heinrich Wölfflin, Principles of Art History: the Problem of the Development of Style in Later Art, M. D. Hottinger (trans.), Mineola, N.Y.: Dover (1950): pp. 18–72.