The A. W. Faber Model 366 was an unusual model of

slide rule The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which is ...

, manufactured in Germany by the A. W. Faber Company around 1909, with scales that followed a system invented by Johannes Schumacher (1858-1930) that used

discrete logarithms In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b'' ...

to calculate products of integers without approximation. The Model 366 is notable for its table of numbers, mapping the numbers 1 to 100 to a permutation of the numbers 0 to 99 in a pattern based on discrete logarithms. The markings on the table are: : The slide rule has two scales on each side of the upper edge of the slider marked with the integers 1 to 100 in a different permuted order, evenly spaced apart. The ordering of the numbers on these scales is :1, 2, 4, 8, 16, 32, 64, 27, 54, 7, 14, 28, 56, 11, 22, 44, 88, 75, 49, 98, 95, 89, 77, 53, 5, 10, 20, 40, 80, 59, 17, 34, 68, 35, 70, 39, 78, 55, 9, 18, 36, 72, 43, 86, 71, 41, 82, 63, 25, 50, 100, 99, 97, 93, 85, 69, 37, 74, 47, 94, 87, 73, 45, 90, 79, 57, 13, 26, 52, 3, 6, 12, 24, 48, 96, 91, 81, 61, 21, 42, 84, 67, 33, 66, 31, 62, 23, 46, 92, 83, 65, 29, 58, 15, 30, 60, 19, 38, 76, 51 which corresponds to the

inverse permutation Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...

to the one given by the number table. There are also two scales on each side of the lower edge of the slider, consisting of the integers 0 to 100 similarly spaced, but in ascending order, with the zero on the lower scales lining up with the 1 on the upper scales. Schumacher's indices are an example of

Jacobi indices In mathematics, the ''Canon arithmeticus'' is a table of indices and powers with respect to primitive roots for prime powers less than 1000, originally published by . The tables were at one time used for arithmetical calculations modulo prime pow ...

, generated with ''p'' = 101 and ''g'' = 2. Schumacher's system of indices correctly generates the desired products, but is not unique: several other similar systems have been created by others, including systems by

Ludgate Ludgate was the westernmost gate in London Wall. Of Roman origin, it was rebuilt several times and finally demolished in 1760. The name survives in Ludgate Hill, an eastward continuation of Fleet Street, Ludgate Circus and Ludgate Square. Ety ...

, Remak and Korn. An elaborate system of rules had to be used to compute products of numbers larger than 101. Very few of the Model 366 slide rules remain, with only five known to have survived.

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Ein Rechenschieber mit Teilung in gleiche Intervalle auf der Grundlage der zahlentheoretischen Indizes. Für den Unterricht konstruiert
(English: "A slide rule with division into equal intervals based on number theoretic indices. Designed for teaching."), Dr. Joh. Schumacher, Munich, 1909 (in German)
''Rechnerlexikon'' article on discrete logarithms, including use in the Schumacher slide rule
(in German)
High resolution images of the Model 366 slide rule
at the

Oughtred Society William Oughtred ( ; 5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernard, T. Birch and J. Lockman, ''A General ...

A Model 366 slide rule made in 1921

Close-up of the number table attached to the cursor
Mechanical calculators Discrete mathematics Multiplication

See also

References

External links