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]

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]

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.

• Faber-Castell Model 366 – Unusual slide rule using a system similar to discrete logarithmsPages displaying short descriptions of redirect targets
• Canon arithmeticus – Book by Carl Jacobi

• 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.

• 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 The origins of digital computers : selected papers. Randell, Brian, 1936-. Berlin: Springer-Verlag. 1973. p. 71. ISBN 978-3-642-96145-8. OCLC 858931618.{{cite book}}: CS1 maint: others (link)
• Method for deriving Ludgate's Irish Logarithms from first principles, Brian Coghlan, 2022)