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.
The following is an implementation of Ludgate's Irish logarithm algorithm in the Python programming language:
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.