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]
See also
- Faber-Castell Model 366 – Unusual slide rule using a system similar to discrete logarithms
- Canon arithmeticus – Book by Carl Jacobi
References
- ↑ Randall, Brian (October 1982). "From Analytical Engine to Electronic Digital Computer:The Contributions of Ludgate, Torres, and Bush" (PDF). Annals of the History of Computing. 4 (4): 20. Archived (PDF) from the original on 2019-12-28. Retrieved 2019-12-28.
- ↑ de Man, Andries. "Irish Logarithms Part 2 – Calculating History". sites.google.com. Archived from the original on 2020-02-23. Retrieved 2019-12-28.
- ↑ de Man, Andries. "Irish Log Animation". Archived from the original on 2020-02-23. Retrieved 2019-12-29.
- 1 2 Coghlan, Brian (2020-06-10). "Percy Ludgate's Logarithmic Indexes" (PDF). treasures.scss.tcd.ie. Retrieved 2023-10-01.
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 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)