Английская Википедия:Irish logarithm
The Irish logarithm was a system of number manipulation invented by Percy Ludgate for machine multiplication. The system used a combination of mechanical cams as lookup tables and mechanical addition to sum pseudo-logarithmic indices to produce partial products, which were then added to produce results.[1]
The technique is similar to Zech logarithms (also known as Jacobi logarithms), but uses a system of indices original to Ludgate.[2]
Concept
Ludgate's algorithm compresses the multiplication of two single decimal numbers into two table lookups (to convert the digits into indices), the addition of the two indices to create a new index which is input to a second lookup table that generates the output product.[3] Because both lookup tables are one-dimensional, and the addition of linear movements is simple to implement mechanically, this allows a less complex mechanism than would be needed to implement a two-dimensional 10×10 multiplication lookup table.
Ludgate stated that he deliberately chose the values in his tables to be as small as he could make them; given this, Ludgate's tables can be simply constructed from first principles, either via pen-and-paper methods, or a systematic search using only a few tens of lines of program code.[4] They do not correspond to either Zech logarithms, Remak indexes or Korn indexes.[4]
Pseudocode
The following is an implementation of Ludgate's Irish logarithm algorithm in the Python programming language:
table1 = [50, 0, 1, 7, 2, 23, 8, 33, 3, 14]
table2 = [ 1, 2, 4, 8, 16, 32, 64, 3, 6, 12,
24, 48, 0, 0, 9, 18, 36, 72, 0, 0,
0, 27, 54, 5, 10, 20, 40, 0, 81, 0,
15, 30, 0, 7, 14, 28, 56, 45, 0, 0,
21, 42, 0, 0, 0, 0, 25, 63, 0, 0,
0, 0, 0, 0, 0, 0, 35, 0, 0, 0,
0, 0, 0, 0, 0, 0, 49, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0]
def product(a: int, b: int) -> int:
"""Ludgate's Irish logarithm algorithm."""
return table2[table1[a] + table1[b]]
Table 1 is taken from Ludgate's original paper; given the first table, the contents of Table 2 can be trivially derived from Table 1 and the definition of the algorithm. Note since that the last third of the second table is entirely zeros, this could be exploited to further simplify a mechanical implementation of the algorithm.
See also
References
Further reading
- Boys, C.V., "A New Analytical Engine," Nature, Vol. 81, No. 2070, July 1, 1904, pp. 14–15.
- Randell, B., "Ludgate's analytical machine of 1909", The Computer Journal, Volume 14, Issue 3, 1971, Pages 317–326, https://doi.org/10.1093/comjnl/14.3.317 Includes the text of Ludgate's original paper.
External links
- A detailed treatment of Ludgate's Irish Logarithms, Brian Coghlan, 2019 (Archived from original link)
- Transcript of "On a Proposed Analytical Machine" by Percy Ludgate (first published in Scientific Proceedings of the Royal Dublin Society 1909 vol 12 pages 77–91), containing Ludgate's own description of the Irish logarithm tables
- A reproduction of Ludgate's original 1909 paper, from Шаблон:Cite book
- Method for deriving Ludgate's Irish Logarithms from first principles, Brian Coghlan, 2022)
