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Balanced ternary

Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2. The balanced ternary system can represent all integers without using a separate minus sign; the value of the leading non-zero digit of a number has the sign of the number itself. The balanced ternary system is an example of a non-standard positional numeral system. It was used in some early computers and has also been used to solve balance puzzles.

Different sources use different glyphs to represent the three digits in balanced ternary. In this article, T (which resembles a ligature of the minus sign and 1) represents −1, while 0 and 1 represent themselves. Other conventions include using '−' and '+' to represent −1 and 1 respectively, or using Greek letter theta (Θ), which resembles a minus sign in a circle, to represent −1. In publications about the Setun computer, −1 is represented as overturned 1: " 1 ".

Balanced ternary makes an early appearance in Michael Stifel's book Arithmetica Integra (1544). It also occurs in the works of Johannes Kepler and Léon Lalanne. Related signed-digit schemes in other bases have been discussed by John Colson, John Leslie, Augustin-Louis Cauchy, and possibly even the ancient Indian Vedas.

Definition
Let $$\mathcal{D}_{3} := \lbrace \operatorname{T}, 0, 1 \rbrace$$ denote the set of symbols (also called glyphs or characters), where the symbol $$\bar{1}$$ is sometimes used in place of $$\operatorname{T}.$$ Define an integer-valued function $$f = f_{\mathcal{D}_{3}} : \mathcal{D}_{3} \to \mathbb{Z}$$ by

 * $$f_{}(\operatorname{T}) = -1,$$
 * $$f_{}(0) = 0 \quad$$ and
 * $$f_{}(1) = 1,$$

where the right hand sides are integers with their usual values. This function, $$f_{},$$ is what rigorously and formally establishes how integer values are assigned to the symbols/glyphs in $$\mathcal{D}_{3}.$$ One benefit of this formalism is that the definition of "the integers" (however they may be defined) is not conflated with any particular system for writing/representing them; in this way, these two distinct (albeit closely related) concepts are kept separate.

The set $$\mathcal{D}_{3}$$ together with the function $$f_{}$$ forms a balanced signed-digit representation called the balanced ternary system. It can be used to represent integers and real numbers.

Ternary integer evaluation
Let $$\mathcal{D}_{3}^{+}$$ be the Kleene plus of $$\mathcal{D}_{3}$$, which is the set of all finite length concatenated strings $$d_n \ldots d_0$$ of one or more symbols (called its digits) where $$n$$ is a non-negative integer and all $$n + 1$$ digits $$d_n, \ldots, d_0$$ are taken from $$\mathcal{D}_{3} = \lbrace \operatorname{T}, 0, 1 \rbrace.$$ The start of $$d_n \ldots d_0$$ is the symbol $$d_0$$ (at the right), its end is $$d_n$$ (at the left), and its length is $$n + 1$$. The ternary evaluation is the function $$v = v_{3} ~:~ \mathcal{D}_{3}^{+} \to \mathbb{Z}$$ defined by assigning to every string $$d_n \ldots d_0 \in \mathcal{D}_{3}^{+}$$ the integer

 * $$v\left( d_n \ldots d_0 \right) ~=~ \sum_{i=0}^{n} f_{} \left( d_{i} \right) 3^{i}.$$

The string $$d_n \ldots d_0$$ represents (with respect to $$v$$) the integer $$v\left( d_n \ldots d_0 \right).$$ The value $$v\left( d_n \ldots d_0 \right)$$ may alternatively be denoted by $${d_n \ldots d_0}_{\operatorname{bal}3}.$$ The map $$v : \mathcal{D}_{3}^{+} \to \mathbb{Z}$$ is surjective but not injective since, for example, $$0 = v(0) = v(00) = v(0 0 0) = \cdots.$$ However, every integer has exactly one representation under $$v$$ that does not end (on the left) with the symbol $$0,$$ i.e. $$d_n = 0 .$$

If $$d_n \ldots d_0 \in \mathcal{D}_{3}^{+}$$ and $$n > 0$$ then $$v$$ satisfies:

 * $$v\left( d_n d_{n-1} \ldots d_0 \right) ~=~ f_{} \left( d_{n} \right) 3^{n} + v\left( d_{n-1} \ldots d_0 \right)$$

which shows that $$v$$ satisfies a sort of recurrence relation. This recurrence relation has the initial condition $$v\left( \varepsilon \right) = 0$$ where $$\varepsilon$$ is the empty string.

This implies that for every string $$d_n \ldots d_0 \in \mathcal{D}_{3}^{+},$$

 * $$v\left( 0 d_n \ldots d_0 \right) = v\left( d_n \ldots d_0 \right)$$

which in words says that leading $$0$$ symbols (to the left in a string with 2 or more symbols) do not affect the resulting value.

The following examples illustrate how some values of $$v$$ can be computed, where (as before) all integer are written in decimal (base 10) and all elements of $$\mathcal{D}_{3}^{+}$$ are just symbols.

 * $$\begin{alignat}{10}

v\left( \operatorname{T} \operatorname{T} \right) &= && f_{}\left( \operatorname{T} \right) 3^{1} + && f_{}\left( \operatorname{T} \right) 3^{0} &&= &&(-1) &&3 &&\,+\, &&(-1) &&1 &&= -4 \\

v\left( \operatorname{T} 1 \right) &= && f_{}\left( \operatorname{T} \right) 3^{1} + && f_{}\left( 1 \right) 3^{0} &&= &&(-1) &&3 &&\,+\, &&(1) &&1 &&= -2 \\

v\left( 1 \operatorname{T} \right) &= && f_{}\left( 1 \right) 3^{1} + && f_{}\left( \operatorname{T} \right) 3^{0} &&= &&(1) &&3 &&\,+\, &&(-1) &&1 &&= 2 \\

v\left( 1 1 \right) &= && f_{}\left( 1 \right) 3^{1} + && f_{}\left( 1 \right) 3^{0} &&= &&(1) &&3 &&\,+\, &&(1) &&1 &&= 4 \\

v\left( 1 \operatorname{T} 0 \right) &= f_{}\left( 1 \right) 3^{2} + && f_{}\left( \operatorname{T} \right) 3^{1} + && f_{}\left( 0 \right) 3^{0} &&= (1) 9 \,+\, &&(-1) &&3 &&\,+\, &&(0) &&1 &&= 6 \\

v\left( 1 0 \operatorname{T} \right) &= f_{}\left( 1 \right) 3^{2} + && f_{}\left( 0 \right) 3^{1} + && f_{}\left( \operatorname{T} \right) 3^{0} &&= (1) 9 \,+\, &&(0) &&3 &&\,+\, &&(-1) &&1 &&= 8 \\ \end{alignat} $$

and using the above recurrence relation

 * $$v\left( 1 0 1 \operatorname{T} \right) = f_{}\left( 1 \right) 3^{3} + v\left( 0 1 \operatorname{T} \right) = (1) 27 + v\left( 1 \operatorname{T} \right) = 27 + 2 = 29.$$

Conversion to decimal
In the balanced ternary system the value of a digit n places left of the radix point is the product of the digit and 3n. This is useful when converting between decimal and balanced ternary. In the following the strings denoting balanced ternary carry the suffix, bal3. For instance,
 * 10bal3 = 1 × 31 + 0 × 30 = 3dec
 * 10𝖳bal3 = 1 × 32 + 0 × 31 + (−1) × 30 = 8dec
 * −9dec = −1 × 32 + 0 × 31 + 0 × 30 = 𝖳00bal3
 * 8dec = 1 × 32 + 0 × 31 + (−1) × 30 = 10𝖳bal3

Similarly, the first place to the right of the radix point holds 3−1 = $1⁄3$, the second place holds 3−2 = $1⁄9$, and so on. For instance,
 * −$2⁄3$dec = −1 + $1⁄3$ = −1 × 30 + 1 × 3−1 = 𝖳.1bal3.

An integer is divisible by three if and only if the digit in the units place is zero.

We may check the parity of a balanced ternary integer by checking the parity of the sum of all trits. This sum has the same parity as the integer itself.

Balanced ternary can also be extended to fractional numbers similar to how decimal numbers are written to the right of the radix point.

 * {| class="wikitable"

! Decimal ! style="text-align: right" | −0.9 ! style="text-align: right" | −0.8 ! style="text-align: right" | −0.7 ! style="text-align: right" | −0.6 ! style="text-align: right" | −0.5 ! style="text-align: right" | −0.4 ! style="text-align: right" | −0.3 ! style="text-align: right" | −0.2 ! style="text-align: right" | −0.1 ! style="text-align: right" | 0 ! Balanced Ternary ! Decimal ! style="text-align: right" | 0.9 ! style="text-align: right" | 0.8 ! style="text-align: right" | 0.7 ! style="text-align: right" | 0.6 ! style="text-align: right" | 0.5 ! style="text-align: right" | 0.4 ! style="text-align: right" | 0.3 ! style="text-align: right" | 0.2 ! style="text-align: right" | 0.1 ! style="text-align: right" | 0 ! Balanced Ternary
 * 𝖳.$\overline{010𝖳}$||𝖳.$\overline{1𝖳𝖳1}$|| 𝖳.$\overline{10𝖳0}$|| 𝖳.$\overline{11𝖳𝖳}$|| 0.$\overline{𝖳}$ or 𝖳.$\overline{1}$ || 0.$\overline{𝖳𝖳11}$ || 0.$\overline{𝖳010}$ || 0.$\overline{𝖳11𝖳}$ || 0.$\overline{0𝖳01}$ || 0
 * 1.$\overline{0𝖳01}$||1.$\overline{𝖳11𝖳}$|| 1.$\overline{𝖳010}$|| 1.$\overline{𝖳𝖳11}$|| 0.$\overline{1}$ or 1.$\overline{𝖳}$ || 0.$\overline{11𝖳𝖳}$ || 0.$\overline{10𝖳0}$ || 0.$\overline{1𝖳𝖳1}$ || 0.$\overline{010𝖳}$ || 0
 * }

In decimal or binary, integer values and terminating fractions have multiple representations. For example, $1⁄10$ = 0.1 = 0.1$\overline{0}$ = 0.0$\overline{9}$. And, $1⁄2$ = 0.12 = 0.1$\overline{0}$2 = 0.0$\overline{1}$2. Some balanced ternary fractions have multiple representations too. For example, $1⁄6$ = 0.1$\overline{𝖳}$bal3 = 0.0$\overline{1}$bal3. Certainly, in the decimal and binary, we may omit the rightmost trailing infinite 0s after the radix point and gain a representations of integer or terminating fraction. But, in balanced ternary, we can't omit the rightmost trailing infinite −1s after the radix point in order to gain a representations of integer or terminating fraction.

Donald Knuth has pointed out that truncation and rounding are the same operation in balanced ternary—they produce exactly the same result (a property shared with other balanced numeral systems). The number $1⁄2$ is not exceptional; it has two equally valid representations, and two equally valid truncations: 0.$\overline{1}$ (round to 0, and truncate to 0) and 1.$\overline{𝖳}$ (round to 1, and truncate to 1). With an odd radix, double rounding is also equivalent to directly rounding to the final precision, unlike with an even radix.

The basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary. Multiplication by two can be done by adding a number to itself, or subtracting itself after a-trit-left-shifting.

An arithmetic shift left of a balanced ternary number is the equivalent of multiplication by a (positive, integral) power of 3; and an arithmetic shift right of a balanced ternary number is the equivalent of division by a (positive, integral) power of 3.

Conversion to and from a fraction
The conversion of a repeating balanced ternary number to a fraction is analogous to converting a repeating decimal. For example (because of 111111bal3 = ($1⁄2$)dec):
 * $$ 0.1\overline{\mathrm{110TT0} } =\tfrac{\mathrm{1110TT0-1} }{\mathrm{111111\times 1T\times 10}}=\tfrac{\mathrm{1110TTT} } {\mathrm{111111\times 1T0}} =\tfrac{\mathrm{111\times 1000T} } {\mathrm{111\times 1001\times 1T0}} =\tfrac{\mathrm{1111\times 1T}}{\mathrm{1001\times 1T0}} =\tfrac{1111}{10010}=\tfrac{\mathrm{1T1T}}{\mathrm{1TTT0}} =\tfrac{101}{\mathrm{1T10} } $$

Irrational numbers
As in any other integer base, algebraic irrationals and transcendental numbers do not terminate or repeat. For example:
 * {| class="wikitable"

!Decimal !! Balanced ternary
 * $$\sqrt{2}=1.4142135623731\ldots$$ || $$\sqrt{\mathrm{1T}}=\mathrm{1.11T1TT00T00T01T0T00T00T01TT\ldots}$$
 * $$\sqrt{3}=1.7320508075689\ldots$$ || $$\sqrt{\mathrm{10}}=\mathrm{1T.T1TT10T0000TT1100T0TTT011T0\ldots}$$
 * $$\sqrt{5}=2.2360679774998\ldots$$ || $$\sqrt{\mathrm{1TT}}=\mathrm{1T.1T0101010TTT1TT11010TTT01T1\ldots}$$
 * $\varphi=\frac{1 + \sqrt{5}}{2}=1.6180339887499\ldots$ || $\varphi=\frac{1 + \sqrt{\mathrm{1TT}}}{\mathrm{1T}}=\mathrm{1T.T0TT01TT0T10TT11T0011T10011\ldots}$
 * $$\tau=6.28318530717959\ldots$$ || $$\tau=\mathrm{1T0.10TT0T1100T110TT0T1TT000001}\ldots$$
 * $$\pi=3.14159265358979\ldots$$ || $$\pi=\mathrm{10.011T111T000T011T1101T111111}\ldots$$
 * $$e=2.71828182845905\ldots$$ || $$e=\mathrm{10.T0111TT0T0T111T0111T000T11T}\ldots$$
 * }
 * $$\tau=6.28318530717959\ldots$$ || $$\tau=\mathrm{1T0.10TT0T1100T110TT0T1TT000001}\ldots$$
 * $$\pi=3.14159265358979\ldots$$ || $$\pi=\mathrm{10.011T111T000T011T1101T111111}\ldots$$
 * $$e=2.71828182845905\ldots$$ || $$e=\mathrm{10.T0111TT0T0T111T0111T000T11T}\ldots$$
 * }
 * $$e=2.71828182845905\ldots$$ || $$e=\mathrm{10.T0111TT0T0T111T0111T000T11T}\ldots$$
 * }
 * }

The balanced ternary expansions of $$\pi$$ is given in OEIS as, that of $$e$$ in.

Conversion from ternary
Unbalanced ternary can be converted to balanced ternary notation in two ways:
 * Add 1 trit-by-trit from the first non-zero trit with carry, and then subtract 1 trit-by-trit from the same trit without borrow. For example,
 * 0213 + 113 = 1023, 1023 − 113 = 1T1bal3 = 7dec.
 * If a 2 is present in ternary, turn it into 1T. For example,
 * 02123 = 0010bal3 + 1T00bal3 + 001Tbal3 = 10TTbal3 = 23dec

If the three values of ternary logic are false, unknown and true, and these are mapped to balanced ternary as T, 0 and 1 and to conventional unsigned ternary values as 0, 1 and 2, then balanced ternary can be viewed as a biased number system analogous to the offset binary system. If the ternary number has n trits, then the bias b is
 * $$b=\left\lfloor \frac{3^n}{2} \right\rfloor$$

which is represented as all ones in either conventional or biased form.

As a result, if these two representations are used for balanced and unsigned ternary numbers, an unsigned n-trit positive ternary value can be converted to balanced form by adding the bias b and a positive balanced number can be converted to unsigned form by subtracting the bias b. Furthermore, if x and y are balanced numbers, their balanced sum is x + y − b when computed using conventional unsigned ternary arithmetic. Similarly, if x and y are conventional unsigned ternary numbers, their sum is x + y + b when computed using balanced ternary arithmetic.

Conversion to balanced ternary from any integer base
We may convert to balanced ternary with the following formula:

\left(a_na_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots\right)_b = \sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}. $$

where,
 * anan−1...a1a0.c1c2c3... is the original representation in the original numeral system.
 * b is the original radix. b is 10 if converting from decimal.
 * ak and ck are the digits k places to the left and right of the radix point respectively.

For instance, −25.4dec = −(1T×1011 + 1TT×1010 + 11×101−1) = −(1T×101 + 1TT + 11÷101) = −10T1.$\overline{1}$ = T01T.$\overline{𝖳}$

1010.12 = 1T10 + 1T1 + 1T−1 = 10T + 1T + 0.$1⁄3$ = 101.$1⁄4$

Addition, subtraction and multiplication and division
The single-trit addition, subtraction, multiplication and division tables are shown below. For subtraction and division, which are not commutative, the first operand is given to the left of the table, while the second is given at the top. For instance, the answer to 1 − T = 1T is found in the bottom left corner of the subtraction table.

|+ Addition |- align="right" ! + !! T !! 0 !! 1 |- |- ! T | T1 || T || 0 |- ! 0 | T || 0 || 1 |- ! 1 | 0 || 1 || 1T |} :{| class="wikitable" style="width: 8em; text-align: center;" |+ Subtraction |- align="right" ! − !! T !! 0 !! 1 |- ! T | 0 || T || T1 |- ! 0 | 1 || 0 || T |- ! 1 | 1T || 1 || 0 |} :{| class="wikitable" style="width: 8em; text-align: center;" |+ Multiplication |- align="right" ! × !! T !! 0 !! 1 |- |- ! T | 1 || 0 || T |- ! 0 | 0 || 0 || 0 |- ! 1  | T || 0 || 1 |} :{| class="wikitable" style="text-align: center;" |+ Division |- align="right" ! ÷ !! T !! 1 |- |- ! T | 1 || T |- ! 0 | 0 || 0 |- ! 1 | T || 1 |}
 * }

Multi-trit addition and subtraction
Multi-trit addition and subtraction is analogous to that of binary and decimal. Add and subtract trit by trit, and add the carry appropriately. For example:

1TT1TT.1TT1             1TT1TT.1TT1            1TT1TT.1TT1          1TT1TT.1TT1 +  11T1.T                −  11T1.T              −  11T1.T     →     +   TT1T.1 ______________         ______________                               _______________            1T0T10.0TT1              1T1001.TTT1                                 1T1001.TTT1 +  1T                   +  T  T1                                    + T  T          ______________         ________________                             ________________ 1T1110.0TT1             1110TT.TTT1                                 1110TT.TTT1 +  T                    + T   1                                     + T   1 ______________        ________________                             ________________            1T0110.0TT1               1100T.TTT1                                  1100T.TTT1

Multi-trit multiplication
Multi-trit multiplication is analogous to that of binary and decimal.

1TT1.TT   ×   T11T.1 _____________        1TT.1TT multiply 1 T11T.11 multiply T       1TT1T.T   multiply 1 1TT1TT    multiply 1 T11T11     multiply T    _____________ 0T0000T.10T

Multi-trit division
Balanced ternary division is analogous to that of binary and decimal.

However, 0.5dec = 0.1111...bal3 or 1.TTTT...bal3. If the dividend over the plus or minus half divisor, the trit of the quotient must be 1 or T. If the dividend is between the plus and minus of half the divisor, the trit of the quotient is 0. The magnitude of the dividend must be compared with that of half the divisor before setting the quotient trit. For example,

1TT1.TT     quotient 0.5 × divisor T01.0 _____________ divisor T11T.1 ) T0000T.10T    dividend                                    T11T1                        T000 < T010, set 1                      _______                        1T1T0                        1TT1T                      1T1T0 > 10T0, set T                        _______                          111T                         1TT1T                      111T > 10T0, set T                        _______                           T00.1                          T11T.1                    T001 < T010, set 1                         ________                           1T1.00                           1TT.1T                  1T100 > 10T0, set T                          ________                            1T.T1T                            1T.T1T                 1TT1T > 10T0, set T                           ________                                 0 Another example,                            1TTT 0.5 × divisor 1T _______ Divisor 11  )1T01T                   1T = 1T, but 1T.01 > 1T, set 1                           11                          _____                           T10                    T10 < T1, set T                            TT                          ______                            T11                   T11 < T1, set T                             TT                           ______                              TT                   TT < T1, set T                              TT                             ____                               0 Another example,                            101.TTTTTTTTT...                          or 100.111111111...         0.5 × divisor 1T  _________________             divisor  11  )111T                    11 > 1T, set 1 11                         _____                             1                     T1 < 1 < 1T, set 0 ___                            1T                    1T = 1T, trits end, set 1.TTTTTTTTT... or 0.111111111...

Square roots and cube roots
The process of extracting the square root in balanced ternary is analogous to that in decimal or binary.
 * $$(10\cdot x+y)^{\mathrm{1T}}-100\cdot x^{\mathrm{1T}}=\mathrm{1T0}\cdot x\cdot y+y^{\mathrm{1T}}=

\begin{cases} \mathrm{T10}\cdot x+1, & y=\mathrm{T} \\ 0, & y=0 \\ \mathrm{1T0}\cdot x+1, & y=1 \end{cases} $$

As in division, we should check the value of half the divisor first. For example, 1. 1 1 T 1 T T 0 0 ...                            _________________________ √ 1T                         1<1T<11, set 1 − 1                            _____                   1×10=10    1.0T                       1.0T>0.10, set 1 1T0  −1.T0                             ________ 11×10=110   1T0T                     1T0T>110, set 1 10T0  −10T0 ________                 111×10=1110    T1T0T                   T1T0T111T0, set 1 10T110  −10T110 __________               111T1×10=111T10    TT1TT0T               TT1TT0T<TTT1T0, set T                        100TTT0   −T001110 ___________              111T1T×10=111T1T0    T001TT0T             T001TT0T<TTT1T10, set T                        10T11110   −T01TTTT0 ____________               111T1TT×10=111T1TT0    T001T0T           TTT1T110<T001T0T<111T1TT0, set 0 −     T           Return 1 ___________             111T1TT0×10=111T1TT00    T001T000T         TTT1T1100<T001T000T<111T1TT00, set 0 −       T         Return 1 _____________           111T1TT00*10=111T1TT000    T001T00000T ... Extraction of the cube root in balanced ternary is similarly analogous to extraction in decimal or binary:
 * $$(10\cdot x+y)^{10}-1000\cdot x^{10}=y^{10}+1000\cdot x^{\mathrm{1T}}\cdot y+ 100\cdot x\cdot y^{\mathrm{1T}}=

\begin{cases} \mathrm{T}+\mathrm{T000}\cdot x^{\mathrm{1T}}+100\cdot x, & y=\mathrm{T}\\ 0, & y=0\\ 1+1000\cdot x^{\mathrm{1T}}+100\cdot x, & y=1 \end{cases}

$$ Like division, we should check the value of half the divisor first too. For example: 1. 1   T  1  0 ...                              _____________________ ³√ 1T − 1                1<1T<10T,set 1 _______                              1.000               1×100=100      −0.100             borrow 100×, do division _______                      1TT     1.T00             1T00>1TT, set 1 1×1×1000+1=1001   −1.001                              __________                                 T0T000 11×100           −   1100           borrow 100×, do division _________                     10T000     TT1T00           TT1T001T1T01TT, set 1 11T×11T×1000+1=11111001   − 11111001 ______________                                    1T10T000 11T1×100                −  11T100      borrow 100×, do division __________                      10T0T01TT     1T0T0T00      T01010T11<1T0T0T00<10T0T01TT, set 0 11T1×11T1×1000+1=1TT1T11001   −  TT1T00      return 100× _____________                                    1T10T000000 ... Hence $\overline{1𝖳}$ = 1.259921dec = 1.1T1 000 111 001 T01 00T 1T1 T10 111bal3.

In computer design
In the early days of computing, a few experimental Soviet computers were built with balanced ternary instead of binary, the most famous being the Setun, built by Nikolay Brusentsov and Sergei Sobolev. The notation has a number of computational advantages over traditional binary and ternary. Particularly, the plus–minus consistency cuts down the carry rate in multi-digit multiplication, and the rounding–truncation equivalence cuts down the carry rate in rounding on fractions. In balanced ternary, the one-digit multiplication table remains one-digit and has no carry and the addition table has only two carries out of nine entries, compared to unbalanced ternary with one and three respectively. Knuth wrote that "Perhaps the symmetric properties and simple arithmetic of this number system will prove to be quite important some day," noting that,

"The complexity of arithmetic circuitry for balanced ternary arithmetic is not much greater than it is for the binary system, and a given number requires only $\log_3 2 \approx 63 \%$ as many digit positions for its representation.""

Other applications
The theorem that every integer has a unique representation in balanced ternary was used by Leonhard Euler to justify the identity of formal power series
 * $$\prod_{n=0}^{\infty} \left(x^{-3^n}+1+x^{3^n}\right)=\sum_{n=-\infty}^{\infty}x^n.$$

Balanced ternary has other applications besides computing. For example, a classical two-pan balance, with one weight for each power of 3, can weigh relatively heavy objects accurately with a small number of weights, by moving weights between the two pans and the table. For example, with weights for each power of 3 through 81, a 60-gram object (60dec = 1T1T0bal3) will be balanced perfectly with an 81 gram weight in the other pan, the 27 gram weight in its own pan, the 9 gram weight in the other pan, the 3 gram weight in its own pan, and the 1 gram weight set aside.

Similarly, consider a currency system with coins worth 1¤, 3¤, 9¤, 27¤, 81¤. If the buyer and the seller each have only one of each kind of coin, any transaction up to 121¤ is possible. For example, if the price is 7¤ (7dec = 1T1bal3), the buyer pays 1¤ + 9¤ and receives 3¤ in change.

They may also provide a more natural representation for the qutrit and systems that use it.

title: Balanced ternary


text:
William Archer (critic)

William Archer (23 September 1856 – 27 December 1924) was a Scottish author, theatre critic, and English spelling reformer based, for most of his career, in London. He was an early advocate of the plays of Henrik Ibsen, and a friend and advocate of George Bernard Shaw.

Life and career
Archer was born in Perth, the eldest boy of the nine children of Thomas Archer and his wife Grace, née Morrison. Thomas relocated frequently seeking employment, and William attended schools in Perth, Lymington, Reigate and Edinburgh. He spent parts of his boyhood with relatives in Norway where he became fluent in Norwegian and became acquainted with the literature of Henrik Ibsen.

Archer won a bursary to the University of Edinburgh to study English literature, moral and natural philosophy, and mathematics. When the family relocated to Australia in 1872, he remained in Scotland as a student. While still at the university he became a leader-writer for the Edinburgh Evening News in 1875, and after a year visiting his family in Australia, he returned to Edinburgh. In 1878, in accordance with his father's wishes, he relocated to London to train as a barrister. He was uninterested in law, and was by now fascinated with theatre, but he entered the Middle Temple and became a barrister in 1883: he never practised. He supported himself by working as dramatic critic of The London Figaro, and after he finished his legal studies he began working for The World, which he continued from 1884 to 1906. In London he soon became known well for his work.

Archer played an important part in introducing Ibsen to the English public, starting with his translation of The Pillars of Society, produced at the Gaiety Theatre in 1880. It was the first play by Ibsen to be produced in London but was little known. He also translated, alone or in collaboration, other productions of Scandinavian drama: Ibsen's A Doll's House (1889), The Master Builder (1893, with Edmund Gosse); Edvard Brandes's A Visit (1892); Ibsen's Peer Gynt (1892, with Charles Archer); Georg Brandes "William Shakespeare"; (1895) Little Eyolf (1895); and John Gabriel Borkman (1897); and he edited Ibsen's Prose Dramas (1890–1891).

In 1881, Archer met Frances Elizabeth Trickett (1855–1929), the youngest of the eight children of John Trickett, a retired engineer. They married in October 1884; the next year they had their only child, Tom (1885–1918), who was killed in action in the First World War. The marriage was enduring and companionable, although Archer began a relationship in 1891, which lasted for the rest of his life, with the actress Elizabeth Robins.

In 1897, Archer, along with Robins, Henry William Massingham, and Alfred Sutro, formed the Provisional Committee to organise an association to produce plays they considered to be of high literary merit, such as Ibsen's. The association was named the "New Century Theatre" but was a disappointment by 1899, although it continued until at least 1904. In 1899, a more successful association, named the Stage Society, was formed to replace it.

Archer was an early friend of George Bernard Shaw, and arranged for his plays to be translated into German. An attempted collaboration on a play failed, although Shaw later used their joint ideas for his early work, Widower's Houses. By Archer's influence Shaw obtained the post of art critic to the periodical The World, before becoming its music critic. A biographer, J. P. Wearing, says of their relationship: "Their intimate friendship could also be very turbulent, since both men were forthright and honest. Shaw respected Archer's intelligence and integrity, and penetrated his formality and deliberately cultivated dour Scots façade. Archer thought Shaw brilliant if perverse, and concluded that he never achieved his great potential because he was too much a jester."

During the First World War, Archer worked for the official War Propaganda Bureau. After the war, he achieved financial success with his play The Green Goddess, produced by Winthrop Ames at the Booth Theatre in New York City in 1921. It was a melodrama, and a popular success, although, he admitted, of much less importance for the art of the drama than his critical work.

Archer died in a London nursing home in 1924 of post-operative complications after the removal of a kidney tumour. Reviewing his life and career, Wearing's summary is that Archer was "a clear, logical man whom some saw as too narrowly rationalistic", but who was perceptive, intuitive and imaginative. Wearing attributes Archer's great influence as a critic to these qualities and to the length of time for which he was engaged in the theatre and reviewing, although "[he] had his blind spots, as in his failure to understand Chekhov, Strindberg, and Shaw, but he was incorruptibly honest and unwaveringly committed to the improvement of ... the theatre. His pioneering advocacy of Ibsen in England cannot be underestimated ... although his other contributions to the theatre are equally valuable."

Archer and Walter Ripman compiled the first dictionary for the English spelling reform system NuSpelling, which would assist the development of SoundSpel.

Critical works

 * English Dramatists of To-day (1882)
 * Henry Irving, a study (1883)
 * About The Theatre: Essays and Studies (1886)
 * Masks or Faces? A Study in the Psychology of Acting (1888)
 * W. C. Macready, a biography (1890)
 * Alan's Wife; a Dramatic Study in Three Scenes (1893)
 * "The Theatrical World for..." (1893–97), in five volumes
 * America To-day, Observations and Reflections (1900)
 * Poets of the Younger Generation (1901) John Lane, the Bodley Head, London
 * Real Conversations (1904)
 * A National Theatre: Scheme and Estimates, with H. Granville Barker, (1907)
 * Through Afro-America (1910)
 * The Life, Trial, and Death of Francisco Ferrer (1911)
 * Play-Making (1912)
 * India and the Future (1917)
 * The Old Drama and the New (1923)

Essays

 * The Great Analysis: A Plea for a Rational World-Order (1912). Introduction by Gilbert Murray

Plays

 * War Is War (1919)
 * The Green Goddess (1921)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: William Archer (critic)


text:
Tōllān

Tollan, Tolan, or Tolán is a name used for the capital cities of two empires of Pre-Columbian Mesoamerica; first for Teotihuacan, and later for the Toltec capital, Tula, both in Mexico. The name has also been applied to the Postclassic Mexican settlement Cholula.

Tollan is a word in Nahuatl used in Spanish colonial documents. Traditionally, tollan was interpreted as a Nahuatl toponym for Tula, Hidalgo, Mexico. More recent scholarship suggests some kind of broader meaning, such as a place of the reeds. Instead of a toponym for a specific settlement, it is also used as a qualifier to denote a category of densely populated cities.

Teotihuacan seems to have been the first city known by this name. After the collapse of the Teotihuacan empire, central Mexico broke into smaller states. The Toltec created the first sizable Mexican empire after the fall of Teotihuacan, and their capital was referred to by the same name as a reference to the earlier greatness of Teotihuacan.

In Aztec accounts at the time of the arrival of the Conquistadores, Teotihuacan and the Toltec capital sometimes seem to be confused and conflated.

The epithet "Tollan" was also sometimes applied to any great metropolis or capital. Cholula, for example, was sometimes called "Tollan Cholula", and the Aztec capital of Tenochtitlán was likewise given the title "Tollan". The Mixtec translation of this, Ñuu Co'yo is still the Mixtec name for Mexico City to this day.

Tollan in Mesoamerican mythology
Tollan is the name given to the mythical place of origin in many Mesoamerican traditions, including those of the Aztecs and the K'iche' Maya. In the K'iche' epic Popol Vuh, the first people created are gathered at Tollan, the place of seven caves, where they receive their languages and their gods.

title: Tōllān


text:
Tolan

Tolan is a surname.

Notable people
Notable people with the surname include:

 * Bobby Tolan (born 1945), American baseball player
 * Brentwood S. Tolan (1855–1923), American architect
 * Eddie Tolan (1908–1967), American athlete and sprinter
 * John H. Tolan (1877–1947), U.S. Representative from California
 * John V. Tolan, historian of religious and cultural relations between the Arab and Latin worlds in the Middle Ages
 * Johnnie Tolan (1917–1986), American racecar driver
 * Michael Tolan (born 1925), American actor
 * Peter Tolan (born 1958), American television producer, director, and screenwriter
 * Stephanie S. Tolan, American author
 * Thomas J. Tolan (1830–1883), American architect
 * Tolan (medieval poet), south Indian poet associated with "kudiyattam"

title: Tolan


text:
Auer von Welsbach

Auer von Welsbach:
 * Alois Auer, Ritter von Welsbach
 * Carl Auer, Freiherr (Baron) von Welsbach, son of Alois

title: Auer von Welsbach


text:
German Federal Coast Guard

The German Federal Coast Guard (Küstenwache des Bundes) is a civilian law enforcement organisation whose primary missions are border protection, maritime environmental protection, shipping safety, fishery protection and customs enforcement. The Küstenwache is an association of several federal agencies, not a single entity like the United States Coast Guard.

The agencies that make up the Küstenwache have a common plan of action and direct their operations from two Coast Guard Centers (German: Küstenwachzentren), Neustadt (Holstein) for the Baltic Sea and Cuxhaven for the North Sea.

Structure and responsibilities

Since 2007 the Joint Situation Center Sea (GLZ-See) is in service. GLZ-See is coordinating the operational units of the federal government (Küstenwache) and the coastal states of Germany (Wasserschutzpolizei).

Units and personnel from these federal agencies make up the coast guard:

 * Bundespolizei (Federal Police), Ministry of the Interior
 * Waterways and Shipping Offices, Federal Waterways and Shipping Administration (WSV), Federal Ministry for Transport, Construction and Urban Development
 * Kontrolleinheit See (Maritime Customs Service), Federal Customs Administration, Federal Ministry of Finance
 * Federal Agency for Agriculture and Nutrition (BLE), Federal Ministry of Consumer Protection, Food, and Agriculture

Coast guard personnel
The personnel serving with the Küstenwache include both police officers (in the case of the Bundespolizei and Wasserzoll) and civilians from the Waterways and Shipping Office and other agencies. Küstenwache personnel do not have combatant status as it is not a military unit as many other coast guards are. In contrast, the United States Coast Guard is both a military service and a law enforcement organisation and its commissioned officers have both police and military powers. However, police officers serving with the Küstenwache retain their usual police powers, adjusted to the maritime nature of their job.

Ships and other vessels
A total of 27 ships and boats from the aforementioned institutions are in operation (Bundespolizei: six ships; Wasserzoll: eight customs cutters, four customs cruisers; WSV: three multi-purpose ships; BLE: two fishery protection boats, three fishery research ships).

The ships are all marked with the legend "Küstenwache" as well as a black-red-gold signet on the hull and the coast guard's coat-of-arms. However, the hull colour differs from agency to agency: blue for the Bundespolizei, green for the Zoll and black for the WSV and BLE.

A number of helicopters and planes are in use as well.

Some Ships:
 * BP 24	Bad Bramstedt
 * BP 25	Bayreuth
 * BP 26	Eschwege
 * BP 81	Potsdam
 * BP 82	Bamberg
 * BP 83	Bad Düben
 * MS Priwall
 * MS Kalkgrund
 * FSB Seeadler
 * Hamburg
 * SUBS Mellum

title: German Federal Coast Guard


text:
DESQview

DESQview (DV) is a text mode multitasking operating environment developed by Quarterdeck Office Systems which enjoyed modest popularity in the late 1980s and early 1990s. Running on top of DOS, it allows users to run multiple programs concurrently in multiple windows.

Desq
Quarterdeck's predecessor to DESQview was a task switching product called Desq (shipped late April or May 1984 ), which allows users to switch between running programs. Quarterdeck revamped its package, bringing multitasking in, and adding TopView compatibility.

DESQview was released in July 1985, four months before Microsoft released the first version of Windows. It was widely thought to be the first program to bring multitasking and windowing capabilities to DOS; in fact, there was a predecessor, IBM TopView, which shipped March 1985,.

Under DESQview, well-behaved DOS programs can be run concurrently in resizable, overlapping windows (something the first version of Windows cannot do). A simple hideable menu allows cutting and pasting between programs. DESQview provides support for simple editable macros as well. Quarterdeck also developed a set of optional utilities for DESQview, including a notepad and dialer. Later versions allow graphics mode programs to be loaded as well, but only run in full screen mode.

DESQview is not a GUI (Graphical User Interface) operating system. Rather, it is a non-graphical, windowed shell that runs in real mode on top of DOS, although it can run on any Intel 8086- or Intel 80286-based PC. It can also use expanded memory add-ons to work around the 640 KB RAM limit of conventional memory on early PCs. DESQview really came into its own on Intel 80386 machines, which are better at utilizing memory above DOS's limit. However, in either case, it runs in real mode rather than protected mode, meaning that a misbehaving program can still crash the system.

DESQview and QEMM
To make maximum use of extended memory on Intel 80386 processors, by transforming it into expanded memory and upper memory blocks (UMBs) accessible to DESQview and other real-mode programs, Quarterdeck developed a sophisticated memory manager. Owing to the foresight of its marketing manager, Quarterdeck marketed it as a separate product, QEMM-386 (Quarterdeck Expanded Memory Manager 386). It became more popular than DESQview itself, and sold steadily for many years, generating over US$150 million in sales from 1987 through 1994. After the release of the Intel Pentium processor, the 386 in QEMM was dropped. The combination package of DESQview and QEMM-386 was called DESQview 386 (DV386).

With the introduction of the 80386, the memory management features were enhanced to allow the system to shift into protected mode but also allow the addresses to be configured in a virtual 8086 mode so that the extended memory can be mapped into addressing frames and accessible to real-mode programs such as DOS. This allows a 386 to implement the LIM (Lotus, Intel, Microsoft) EMS (expanded memory specification). The memory manager is easily controlled by the user with DOS program QEMM.COM.

DESQview is able to use QEMM's features far beyond just the LIM EMS API, mapping most of the "conventional" address space (below 640 KB) into multiple extended memory blocks such that each can execute transparently during its context. The main copy of DOS and any device and networking drivers have to be loaded before DESQview. The resulting space is the largest single program that can run, but DESQview under QEMM can run as many instances of those programs as the EMS allows. So an 8 MB system can generally have a dozen full-sized DOS programs running concurrently; a 16 MB system can run over twenty, and so on.

DESQview usage
DESQview was noteworthy in that it supported all common DOS-compatible programs and achieved a degree of performance and stability that was remarkable, given the constraints of its host operating system. It also has a clever interface that is generally unobtrusive while being quickly available and very easy to learn.

All normal PCs include a keyboard with three "shift" or "modifier" keys: Control, Alt, and the normal Shift keys. These keys are normally held down in combination with other keys. DESQview, by default, monitors the Alt key for isolated presses (not in combination). Pressing the Alt key by itself brings up the DESQview menu allowing access to the program's features: start new tasks, switch among them, mark text on the screen, paste text as input into the current task, resize or move the text windows, configure new menu items, etc. In addition a Shift+Alt combination causes DESQview to learn a set of keys as a macro. This allows DESQview to run other programs without interfering with any of the "keybindings" they might be using.

DESQview was critically acclaimed and won many fans, but it never met with mass appeal, despite Quarterdeck's sustained efforts to win people over.

In one area, however, DESQview was a lasting success: many multiuser bulletin board systems were based on it, thanks to its modest hardware requirements, robust multitasking, and superlative handling of multiple communication ports. Most free or inexpensive BBS software of the time ran as a single-node, single-tasking DOS program. Normally, only one copy of the BBS software could run at once, limiting the host PC to running one node. DESQview allows multiple copies of these single-task programs to run at once on the same computer, allowing anyone with even unimpressive hardware to run a large, powerful, multiuser BBS out of their home.

Decline of DESQview
DESQview does not provide a graphical user interface (GUI). While Quarterdeck did provide suites of programming libraries and utilities to support the development of software to use its features these never became widely popular. DESQview's ability to run most software with no modification and the cost of "run-time" licenses, combined with the costs of the development suites themselves made this an unreasonable combination for commercial shrink-wrapped software publishers and vendors.

Microsoft released Windows 3.0 with its own memory management and multitasking features. While DESQview was far faster, smaller, and more stable, it was more expensive and didn't include support for the graphical features of MS Windows.

The decline of QEMM started with the bundling of a memory manager in Digital Research's DR DOS 5.0, released in 1990. To catch on, Microsoft included its own EMM386 in MS-DOS 5.0, while previously the memory management functionality was only available with Windows. QEMM could still be used instead, notably with Windows 3.1x, but only for incremental benefit. Sales of QEMM declined. In August 1994, after three quarters of losses, the company laid off 25% of their employees and the CEO, president, and founder Terry Myers resigned.

As users moved from DESQview to other platforms, notably Windows 3.x and OS/2, third party utility authors wrote utility programs that emulated some DESQview API functions to allow suitably equipped DOS programs to co-operate with these OS. The most notable are TAME (for Windows) and OS/2SPEED (for OS/2).

DESQview/X
Quarterdeck eventually also released a product named DESQview/X (DVX), which is an X Window System server running under DOS and DESQview and thus provides a GUI to which X software (mostly Unix) could be ported.

DESQview/X had three window managers that it launched with, X/Motif, OPEN LOOK, and twm. The default package contained only twm, the others were costly optional extras, as was the ability to interact on TCP/IP networks. Mosaic was ported to DVX.

DVX itself can serve DOS programs and the 16-bit Windows environment across the network as X programs, which made it useful for those who wished to run DOS and Windows programs from their Unix workstations. The same functionality is available with some versions of NCD Wincenter.

NetWare Access Server
Internetworking company Novell developed a product called NetWare Access Server (NAS) incorporating features of DESQview 386 and pcAnywhere. The DESQview multitasking support was used to create an environment into which up to 16 PC and Macintosh clients can login remotely to access NetWare services and run DOS applications.

DESQview after X
DESQview development continued in parallel with DESQview/X. After ceasing development on DESQview/X, another version of DESQview was released. QEMM was still developed after the discontinuation of DESQview, and a version compatible with Windows 98 was released.

In the mid-1990s, Quarterdeck tried to recast itself as an Internet company, releasing a version of the Mosaic web browser. Eventually, the company was acquired by Symantec.

Reception
BYTE in 1984 stated that DesQ's application compatibility was the highest of the five window managers tested. The magazine's Jerry Pournelle in 1989 noted that "while MultiFinder doesn't work very well yet, DESQView on a big 80386 machine certainly does". The magazine that year listed version 3.0 as among the "Distinction" winners of the BYTE Awards, stating that "unlike OS/2, DESQview lets you run the programs you've already paid for ... Many users will find that DESQview is all they need".

title: DESQview


text:
Flex

Flex or FLEX may refer to:

Computing

 * Flex (language), developed by Alan Kay
 * FLEX (operating system), a single-tasking operating system for the Motorola 6800
 * FlexOS, an operating system developed by Digital Research
 * FLEX (protocol), a communication protocol for pagers
 * Flex, a family of automatic test equipment produced by Teradyne
 * FLEx, a piece of software used in language documentation
 * Apache Flex, formerly Adobe Flex, technologies for developing rich internet applications
 * Flex (lexical analyser generator), a lexical analyser generator and a free software alternative to lex
 * Flex machine, a computer developed by RSRE in the 1980s
 * CSS Flexible Box Layout, commonly known as Flexbox, a CSS 3 web layout model

Science and technology

 * Bending, also known as flexure, as used in mechanics
 * Flexion, in anatomy, a position made possible by the joint angle decreasing
 * Femtosecond Lenticule EXtraction, a form of refractive eye surgery
 * Flex circuit, a flexible printed circuit used in electronic assemblies
 * FLEX mission, a future satellite launch mission by the European Space Agency
 * Flex temp, a reduced takeoff thrust setting which trades excess performance for reduced engine wear
 * Flex-fuel, a flexible-fuel vehicle
 * Inflection point, of a curve in geometry
 * Power cord, or flex, a flexible electrical cable

Music

 * Flex (singer), Félix Danilo Gómez (born 1974), Panamanian singer
 * Funkmaster Flex (born 1968), American DJ and music producer
 * Flex (album), Lene Lovich's 1979 second album
 * Flex (EP) a 2003 EP by Pitch Black
 * "Flex" (Dizzee Rascal song), 2007
 * "Flex" (Mad Cobra song), 1992
 * "Flex" (Polo G song), 2020
 * "Flex (Ooh, Ooh, Ooh)", a 2015 song by Rich Homie Quan
 * "Flex", a 2021 song by TFN
 * Flex, a break in the recitation tone before the mediation in Gregorian chant
 * A "flex" in a "flex bar", a technique used in battle rap

Organizations

 * Flex (club), a nightclub in Vienna, Austria
 * Flex (company) (NASDAQ symbol: FLEX), a contract electronics maker based in Singapore
 * Flex FM, a London-based radio station
 * Flex-Elektrowerkzeuge, a German producer of power tools
 * Flex Linhas Aéreas, a Brazilian regional airline

People

 * Flex Alexander, Mark Alexander Knox (b. 1970), an American actor and comedian
 * Walter Flex (1887–1917), German author

Other uses

 * Amazon Flex, a platform for gig workers performing deliveries for Amazon
 * Flex (comics), a fictional superhero
 * Flex (film), a commercial/short film by English director Chris Cunningham
 * Flex (magazine), an American bodybuilding magazine
 * Flex (TV program), a 2021 Philippine variety show
 * Fleet Landing Exercises, a series of landing exercises conducted by the Fleet Marine Force, a combined-United States Navy/Marine landing force
 * Flex Your Rights, a civil-liberties non-profit based in Washington DC, US
 * Ford Flex, a crossover SUV produced from 2009-2019
 * Future Leaders Exchange, an American student exchange program for high school students from the former Soviet Union

title: Flex


text:
Jonathon Prandi

Jonathon Prandi (born August 11, 1972) is an American male fashion model, actor and IT-consultant. He was born in Kalamazoo, Michigan and grew up in Miami, Florida. Jonathon Prandi has been modeling for fashion shows, catalogs and magazines and has appeared in training videos and fitness- and training magazines such as Men's Health, Men's Workout, and Exercise for Men.

He has also appeared several times in Playgirl Magazine, the first time in February 1996. In the September 1996 issue he was their Centerfold, and in 1997 he was Playgirl Magazine's "Man of the Year."

He has appeared in TV shows like Guiding Light, Entertainment Tonight and Extra, and made guest appearances on Fox After Breakfast, Good Day New York, and The Montel Williams Show.

Prandi has also appeared on stage and in films; for example, he played the villain in the Off Broadway Play The Three Musketeers, and he also played himself in the independent film Three's A Crowd. He had a role in Deadly Ties as well as in the independent film Passion Fruits. Jonathon also was Pierce Brosnan's body double in the film The Thomas Crown Affair in 1999.

Prandi also holds a B.S. degree in information technology and an A.S. degree in computer information systems, and has in recent years started a successful web design company.

Prandi (sometimes referred to as Jonathan Prandi or "Sparkles") is 6 ft tall and weighs 200 lb. According to his Playgirl interview, Jonathan is of Italian-American and Native-American ancestry

Filmography

 * Little Shop of Erotica
 * Guiding Light
 * Entertainment Tonight
 * Extra
 * Fox After Breakfast
 * Good Day New York
 * The Montel Williams Show
 * Three's A Crowd
 * Deadly Ties
 * Passion Fruits
 * The Thomas Crown Affair

Stage Appearances

 * There's Something About Sparkles (performed Spring 2019)

title: Jonathon Prandi


text:
Étienne Méhul

Étienne Nicolas Méhul (22 June 1763 – 18 October 1817) was a French composer of the late classical and early romantic periods. He was known as "the most important opera composer in France during the Revolution". He was also the first composer to be called a "Romantic". He is known particularly for his operas, written in keeping with the reforms introduced by Christoph Willibald Gluck and Wolfgang Amadeus Mozart.

Life
Méhul was born at Givet in Ardennes to Jean-François Méhul, a wine merchant, and his wife Marie-Cécile (née Keuly). His first music lessons came from a blind local organist. When he showed promise, he was sent to study with a German musician and organist, Wilhelm Hanser, at the monastery of Lavaldieu, a few miles from Givet. Here Méhul developed his lifelong love of flowers.

In 1778 or 1779 he went to Paris and began to study with Jean-Frédéric Edelmann, a harpsichord player and friend of Méhul's idol Christoph Willibald Gluck. Méhul's first published composition was a book of piano pieces in 1783. He also arranged airs from popular operas and by the late 1780s he had begun to think about an operatic career for himself.

In 1787, the writer Valadier offered Méhul one of his libretti, Cora, which had been rejected by Gluck in 1785. The Académie royale de musique (the Paris Opéra) put Méhul's work, under the title Alonzo et Cora, into rehearsal in June 1789. However, the rehearsals were abandoned on 8 August, probably because the Opéra had been suffering severe financial difficulties throughout the 1780s, and the opera was not premiered until 1791. In the meantime, Méhul found an ideal collaborator in the librettist François-Benoît Hoffman, who provided the words to the first of Méhul's operas to be performed, Euphrosine. Its premiere in 1790 was an immense success and marked the composer out as a new talent. It was also the start of his long relationship with the Comédie Italienne theatre (soon to be renamed the Opéra-Comique).

In spite of the failure of Cora in 1791 and the banning of Adrien for political reasons the year after that, Méhul consolidated his reputation with works such as Stratonice and Mélidore et Phrosine. During the French Revolution, Méhul composed many patriotic songs and propaganda pieces, the most famous of which is the Chant du départ. Méhul was rewarded by becoming the first composer named to the newly founded Institut de France in 1795. He also held a post as one of the five inspectors of the Conservatoire de Paris. Méhul was on friendly terms with Napoleon and became one of the first Frenchmen to receive the Légion d'honneur.

Méhul's operatic success was not as great in the first decade of the nineteenth century as it had been in the 1790s, although works such as Joseph (1807) became famous abroad, particularly in Germany. The failure of his opera Les amazones in 1811 was a severe blow and virtually ended his career as a composer for the theatre. Despite his friendship with Napoleon, Méhul's public standing survived the transition to the Bourbon Restoration intact. However, the composer was now seriously ill with tuberculosis and he died on 18 October 1817. His grave is at Père Lachaise Cemetery, near the grave of the composer François Joseph Gossec.

In 1797 Méhul adopted his seven-year-old nephew, composer Joseph Daussoigne-Méhul, and Joseph's younger brother. He played a major role in his nephew's musical education and career; counting him among his pupils at the Conservatoire de Paris. After his death, Daussoigne-Méhul completed Méhul's unfinished opera Valentine de Milan which premiered at the Opéra-Comique in 1822. He also wrote new recitatives for his opera Stratonice in 1821 for a revival of that work in Paris.

Operas
Méhul's most important contribution to music was his operas. He led the generation of composers who emerged in France in the 1790s, which included his friend and rival Luigi Cherubini and his outright enemy Jean-François Le Sueur. Méhul followed the example of the operas which Gluck had written for Paris in the 1770s and applied Gluck's "reforms" to opéra comique (a genre which mixed music with spoken dialogue and was not necessarily at all "comic" in mood). But he pushed music in a more Romantic direction, showing an increased use of dissonance and an interest in psychological states such as anger and jealousy, thus foreshadowing later Romantic composers such as Weber and Berlioz. Indeed, Méhul was the first composer to be styled a Romantic; a critic used the term in La chronique de Paris on 1 April 1793 when reviewing Méhul's Le jeune sage et le vieux fou.

Méhul's main musical concern was that everything should serve to increase the dramatic impact. As his admirer Berlioz wrote:"[Méhul] was fully convinced that in truly dramatic music, when the importance of the situation deserves the sacrifice, the composer should not hesitate as between a pretty musical effect that is foreign to the scenic or dramatic character, and a series of accents that are true but do not yield any surface pleasure. He was convinced that musical expressiveness is a lovely flower, delicate and rare, of exquisite fragrance, which does not bloom without culture, and which a breath can wither; that it does not dwell in melody alone, but that everything concurs either to create or destroy it – melody, harmony, modulation, rhythm, instrumentation, the choice of deep or high registers for the voices or instruments, a quick or slow tempo, and the several degrees of volume in the sound emitted."

One way in which Méhul increased dramatic expressivity was to experiment with orchestration. For example, in Uthal, an opera set in the Highlands of Scotland, he eliminated violins from the orchestra, replacing them with the darker sounds of violas in order to add local colour. Méhul's La chasse du jeune Henri (Young Henri's Hunt) provides a more humorous example, with its expanded horn section portraying yelping hounds as well as giving hunting calls. (Sir Thomas Beecham frequently programmed this piece to showcase the Royal Philharmonic horn section.)

Méhul's key works of the 1790s were Euphrosine, Stratonice, Mélidore et Phrosine and Ariodant. Ariodant, though a failure at its premiere in 1799, has come in for particular praise from critics. Elizabeth Bartlet calls it "Mehul's best work of the decade and a highpoint of Revolutionary opera". It deals with the same tale of passion and jealousy as Handel's 1735 opera Ariodante. As in many of his other operas, Mehul makes use of a structural device called the "reminiscence motif", a musical theme associated with a particular character or idea in the opera. This device looks forward to the leitmotifs in Richard Wagner's music dramas. In Ariodant, the reminiscence motif is the cri de fureur ("cry of fury"), expressing the emotion of jealousy.

Around 1800, the popularity of such stormy dramas began to wane, replaced by a fashion for the lighter opéra comiques of composers such as François-Adrien Boieldieu. In addition, Mehul's friend Napoleon told him he preferred a more comic style of opera. As a Corsican, Napoleon's cultural background was Italian, and he loved the opera buffa of composers like Paisiello and Cimarosa. Méhul responded with L'irato ("The Angry Man"), a one-act comedy premiered as the work of the Italian composer "Fiorelli" in 1801. When it became an immediate success, Méhul revealed the hoax he had played. Méhul also continued to compose works in a more serious vein. Joseph, based on the Biblical story of Joseph and his brothers, is the most famous of these later operas, but its success in France was short-lived. In Germany, however, it won many admirers throughout the nineteenth century, including Wagner. A melody from Joseph is very similar to a popular folk melody widely known in Germany which was used as a song in the Imperial German Navy, and adapted, notoriously, as the tune for the co-national anthem of Nazi Germany, the Horst-Wessel-Lied. It is unclear, however, whether Méhul's melody was the actual provenance of the melody.

Symphonies and other works
Besides operas, Méhul composed a number of songs for the festivals of the republic (often commissioned by the emperor Napoleon), cantatas, and five symphonies in the years 1797 and 1808 to 1810.

Mehul's First Symphony (1808) is notable for its dissonant and violent mood, and has been compared to Beethoven's Symphony No. 5, written in the same year. Taking inspiration from the more anguished works of Haydn and Mozart, such as Haydn's Sturm und Drang and later Paris Symphonies of 1785–86 and Mozart's Symphony No. 40 (K. 550, 1788), it was revived in one of Felix Mendelssohn's concerts with the Leipzig Gewandhaus Orchestra in 1838 and 1846 to an audience including Robert Schumann, who was impressed by the piece. (At the time of writing, only Beethoven's Symphonies Nos. 1 and 2 [1799/1800 and 1802] had been performed in France.) His other symphonies also followed German and Austrian models. Commenting after the premiere of his first symphony, he noted: "I understood all the dangers of my enterprise; I foresaw the cautious welcome that the music-lovers would give my symphonies. I plan to write new ones for next winter and shall try to write them… to accustom the public gradually to think that a Frenchman may follow Haydn and Mozart at a distance."

A fifth symphony was never completed—"as disillusionment and tuberculosis took their toll", in the words of David Charlton. The Symphonies Nos. 3 and 4 were only rediscovered by Charlton in 1979. Interviewed 8 November 2010 on the BBC Radio 4's Today programme, Professor Charlton said that Méhul's 4th Symphony was the first ever to employ the cyclical principle.

For piano

 * 3 Sonates for Piano, op. 1 (1783)
 * 3 Sonates for Piano, op. 2 (1788)

Orchestral music

 * Ouverture burlesque (1794)
 * Ouverture pour instruments à vent (1794)
 * Symphony in C (1797, only parts are surviving)
 * Symphony No. 1 in G minor (1808/09)
 * Symphony No. 2 in D major (1808/09)
 * Symphony No. 3 in C major (1809)
 * Symphony No. 4 in E major (1810)
 * Symphony No. 5 (1810, only the first movement survives; incomplete and unperformed)

Vocal music

 * Chant du départ (1794)
 * Chant des victoires (1794)
 * Messe Solennelle pour soli, chœurs et orgue (1804)
 * Chant du retour pour la Grande Armée (1808)
 * Chant lyrique pour l'inauguration de la statue de Napoléon (1811)

Ballets

 * Le jugement de Pâris (1793)
 * La dansomanie (1800)
 * Persée et Andromède (1810) (together with music by Haydn, Paer, and Steibelt)

Incidental music for plays

 * Timoléon (by Marie-Joseph Chénier)
 * Les Hussites (by Alexandre Duval)

Discography

 * Uthal. Karine Deshayes, Yann Beuron, Jean-Sébastien Bou, Christophe Rousset (conductor), Les Talens Lyriques, Chœur de chambre de Namur. Bru Zane, 2017.
 * Adrien. Gabrielle Philiponet, Philippe Do, Marc Barrard, Philippe Talbot, Nicolas Courjal, Jean Teigen, Jennifer Borghi, György Vashegyi (dir.), Orfeo Orchestra, Purcell Choir. Bru Zane, 2014.
 * Joseph. Natalie Dessay, soprano; Brigitte Lafon, mezzo-soprano; Laurence Dale, Antoine Normand, Philippe Pistole, tenors; René Massis, baritone; Frédéric Vassar, Philippe Jorquera, basses; Abbi Patrix, speaker; Ensemble choral "Intermezzo"; Orchestre régional de Picardie "Le Sinfonietta"; Claude Bardon, conductor. Chant du monde, c. 1989.
 * Piano Sonatas opp. 1 (Nos. 1–3) & 2 (Nos. 4–6). Brigitte Haudebourg, piano. Arcobaleno, c. 1990.
 * Chant national du 14 juillet 1800; Hymne à la raison; Le chant du départ. Chœur et Orchestre du Capitole de Toulouse; Michel Plasson, conductor. EMI, 1990.
 * The Complete Symphonies (Symphonies Nos. 1–4; including the Ouvertures La Chasse du jeune Henri and Le Trésor supposé). Lisbon Gulbenkian Foundation Orchestra, Michel Swierczewski. Nimbus Records, 1992.
 * La Chasse du jeune Henri, Le Trésor supposé and Timoléon. Royal Philharmonic Orchestra, Sir Thomas Beecham. Sony, 2002.
 * Stratonice. Petibon, Beuron, Lescoart, Daymond, Corona Coloniensis, Cappella Coloniensis, William Christie, Erato 1996.
 * Symphonies Nos 1 and 2. Les Musiciens du Louvre, Marc Minkowski, Erato/Apex, 2003.
 * Overtures: Mélidore et Phrosine; Ariodant; Joseph; Horatius Coclès; Bion; Le jeune sage et le vieux fou; Le trésor supposé; Les deux aveugles de Tolède; La chasse du jeune Henri. Orchestre de Bretagne, Stefan Sanderling, ASV, 2003.
 * Joseph (as Joseph in Ägypten), two versions of the work in German both recorded in 1955: (a) Alexander Welitsch, Libero di Luca, Horst Guenter, Ursula Zollenkopf, NDR Chor and orchestra of the NWDR, Wilhelm Schüchter; (b) Alexander Welitsch, Josef Traxel, Bernhard Michaelis, Friederike Sailer, Stuttgart Radio Symphony Orchestra, Südfunk-Chor, Alfons Rischner. Gala, 2004.
 * L'Irato. Turk, Auvity, Courtin, Buet, Chamber Choir Bonn, L'arte del mondo, Walter Ehrhardt. Capriccio, 2006.
 * Le chant du départ; Chant Funèbre à la Mémoire de Féraud; Hymne pour la Fête des Epoux; Ouverture. Edwige Perfetti, soprano; Tibère Raffali, Christian Papis, tenors; Gilles Cachemaille, baritone; Orchestre d'Harmonie des Gardiens de la Paix de Paris; Claude Pichaureau, conductor; Chœur de l'Armée Française; Serge Zapolski, chorus-master; Chorale a Chœur Joie la Gondoire; Daniel Catenne, chorus-master; Chorale Populaire de Paris; Jean-Claude Chambard, chorus-master. Musifrance, n.d.

title: Étienne Méhul


text:
Gas mantle

An incandescent gas mantle, gas mantle or Welsbach mantle is a device for generating incandescent bright white light when heated by a flame. The name refers to its original heat source in gas lights which illuminated the streets of Europe and North America in the late 19th century. Mantle refers to the way it hangs like a cloak above the flame. Gas mantles were also used in portable camping lanterns, pressure lanterns and some oil lamps.

Gas mantles are usually sold as a fabric bag which, because of impregnation with metal nitrates, burns away to leave a rigid but fragile mesh of metal oxides when heated during initial use; these metal oxides produce light from the heat of the flame whenever used. Thorium dioxide was commonly a major component; being radioactive, it has led to concerns about the safety of those involved in manufacturing mantles. Normal use, however, poses minimal health risk.

Mechanism

The mantle is a roughly pear-shaped fabric bag, made from silk, ramie-based artificial silk, or rayon. The fibers are impregnated with metallic salts; when the mantle is first heated in a flame, the fibers burn away in seconds and the metallic salts convert to solid oxides, forming a brittle ceramic oxide shell in the shape of the original fabric. A mantle glows brightly in the visible spectrum while emitting little infrared radiation. The rare-earth (cerium) and actinide (thorium) oxides in the mantle have a low emissivity in the infrared (in comparison with an ideal black body) but have high emissivity in the visible spectrum. There is also some evidence that the emission is enhanced by candoluminescence, the emission of light from the combustion products before they reach thermal equilibrium. The combination of these properties yields a mantle that, when heated by a kerosene or liquified petroleum gas flame, emits intense radiation that is mostly visible light, with relatively little energy in the unwanted infrared, increasing the luminous efficiency.

The mantle aids the combustion process by keeping the flame small and contained at higher fuel flow rates than in a simple lamp. Concentrating combustion inside the mantle improves the transfer of heat from the flame to the mantle. The mantle shrinks after all the fabric material has burned away during installation leaving a very fragile ceramic oxide shell after its first use.

History
For centuries, artificial light has been generated using open flames. Limelight was invented in the 1820s, but the temperature required to produce visible light through black-body radiation alone was too high to be practical for small lights. In the late 19th century, several inventors tried to develop an effective alternative based on heating a material to a lower temperature but using the emission of discrete spectral lines to simulate white light.

Many early attempts used platinum-iridium gauze soaked in metal nitrates, but these were not successful because of the high cost of these materials and their poor reliability. The first effective mantle was the Clamond basket in 1881, named after its inventor. This device was made from a matrix of magnesium oxide, which did not need to be supported by a platinum wire cage, and was exhibited in the Crystal Palace exhibition of 1883.

The modern gas mantle was one of the many inventions of Carl Auer von Welsbach, a chemist who studied rare-earth elements in the 1880s and who had been Robert Bunsen's student. Ignaz Kreidl worked with him on his early experiments to create the Welsbach mantle. His first process used a mixture of 60% magnesium oxide, 20% lanthanum oxide and 20% yttrium oxide, which he called "Actinophor" and patented in 1887 (March 15, 1887, US patent #359,524). These original mantles gave off a green-tinted light and were not very successful. Welsbach's first company established a factory in Atzgersdorf in 1887, but it failed in 1889. In 1889, Welsbach received his first patent mentioning thorium (March 5, 1889, US patent #399,174). In 1891 he perfected a new mixture of 99% thorium dioxide and 1% cerium dioxide that gave off a much whiter light and produced a stronger mantle. After introducing this new mantle commercially in 1892, it quickly spread throughout Europe. The gas mantle remained an important part of street lighting until the widespread introduction of electric lighting in the early 1900s.

Production

To produce a mantle, cotton is woven or knit into a net bag, impregnated with soluble nitrates of the chosen metals, and then transported to its destination. The user installs the mantle and then burns it to remove the cotton bag and convert the metal nitrates to nitrites which fuse together to form a solid mesh. As the heating continues, the nitrites finally decompose into a fragile mesh of solid oxides with very high melting points.

Early mantles were sold in the unheated cotton mesh condition, since the post heating oxide structure was too fragile to transport easily. The mantle converts to its working form when the cotton burns away on first use. Originally, unused mantles could not be stored for very long because the cotton quickly rotted due to the corrosive nature of the acidic metal nitrates. The acidic metal corrosion was later addressed by soaking the mantle in an ammonia solution to neutralize the excess acid.

Later mantles were made from guncotton (nitrocellulose) which can be produced with extremely fine threads when compared with ordinary cotton threads. These had to be converted back to cellulose by immersion in ammonium sulfide before first use as guncotton is highly flammable and can be explosive. Later, it was discovered that a cotton mantle could be strengthened sufficiently by dipping it in a solution of collodion to coat it with a thin layer that would be burned off when the mantle was first used.

Mantles have a binding thread to tie them to the lamp fitting. Until asbestos was banned due to its carcinogenicity, an asbestos thread was used. Modern mantles use a wire or a ceramic fiber thread.

Safety concerns
Thorium is radioactive and produces the radioactive gas radon-220 as one of its decay products. Moreover, when heated to incandescence, the thorium volatilizes its in-growth radio-daughters, particularly radium-224. Despite its very short half-life, radium quickly replenishes from its radio-parent (thorium-228), and every new heating of the mantle to incandescence releases a fresh flush of radium-224 into the air. This byproduct can be inhaled if the mantle is used indoors, and is an internal alpha-emitter radio-toxicity concern. Secondary decay products of thorium include radium and actinium. Because of this, there are concerns about the safety of thorium mantles. The Australian Radiation Protection and Nuclear Safety Agency recommends mantles made with yttrium instead.

A study in 1981 estimated that the dose from using a thorium mantle every weekend for a year would be 3 –, tiny in comparison to the normal annual background radiation dose of around 2.4 mSv, although this assumes the thorium remains intact rather than airborne. A person actually ingesting a mantle would receive a dose of 2 mSv. However, the radioactivity is a major concern for people involved with the manufacture of mantles and an issue with contamination of soil around some former factory sites.

One potential cause for concern is that particles from thorium gas mantles "fall out" over time and get into the air, where they may be ingested in food or drink. These particles may also be inhaled and remain in the lungs or liver, causing long-term exposure exceeding the risk of background radiation. Also of concern is the release of thorium-bearing dust if the mantle shatters due to mechanical impact.

All of these issues have led to the use of alternatives in some countries, usually yttrium or sometimes zirconium, although these are usually either more expensive or less efficient. Safety concerns were the subject of a federal suit against the Coleman Company (Wagner v. Coleman), which initially agreed to place warning labels on the mantles for this concern, and subsequently switched to using yttrium.

In June 2001, the U.S. Nuclear Regulatory Commission published a study about the Systematic Radiological Assessment of Exemptions for Source and Byproduct Materials, stating that radioactive gas mantles are explicitly legal in the US.

title: Gas mantle