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My favourite APL symbol

by Roger K.W. Hui

Vector 26:1, page 36

language design mathematics and statistics

⍟ ○*

One of the distinguishing characteristics of APL is its unique character set, containing 150-200 symbols. My favorite is ⍟, the symbol for logarithm. Originally, the log symbol was formed by ‘overstriking’ ○ (circle) and * (exponential or power). At present, ⍟ is Unicode[1] code point 0x235F.

Reasons for liking ⍟

  • It’s kind of cute, possessing a radial symmetry.
  • It denotes a function for which conventional mathematical notation [2] does not have a good symbol: 
            ⍟y   ←→ ln y or log y

        x⍟y ←→ logx y
  • It alludes to 0=1+*○0j1, the most beautiful equation in all of mathematics [3], relating in one short phrase the fundamental quantities 0, 1, e, π, and 0j1 and the basic operations plus, times, and exponentiation.
  • It is a visual pun – the symbol looks like the cross-section of a felled tree, i.e. a log [4].

Chronology

1962-03
In A Programming Language [5], logarithm, exponential, and power were not assigned symbols.
1966-03
In Elementary Functions[6], exponential and power were denoted *y and x*y, their definitions to this day. Natural logarithm was denoted *' and base-x logarithm was denoted (x*)'. (In the book, f' is the inverse of f.)
1966-11-27 15:53:58 (GMT-7)
Initial implementation of APL\360 [7].
1967-10-17
Natural logarithm was denoted by ⍟y no later than the publication of The APL\360 Terminal System [8]. The dyadic case x⍟y, base-x log of y, was undefined; instead, it was computed by a defined function in the public library workspace 1 utility [9].
1968-08
Finally, natural logarithm was denoted ⍟y and the base-x logarithm of y was denoted x⍟y, their definitions to this day, no later than the publication of APL\360 User’s Manual [10].

References

  1. Unicode Consortium, Unicode Standard 6.2, 2013
    www.unicode.org/charts/PDF/U2300.pdf
  2. Abramowitz, Milton, and Irene A. Stegun, Handbook of Mathematical Functions, US National Bureau of Standards, 1964; Chapter 4
    people.math.sfu.ca/~cbm/aands/page_67.htm,
  3. Hui, Roger K.W., Euler’s Identity, J Wiki Essay, 2010-02-04
    www.jsoftware.com/jwiki/Essays/Euler's_Identity
  4. McDonnell, Eugene E., The Story of ○, APL Quote-Quad,
    Volume 8, Number 2, 1977-12 www.jsoftware.com/papers/eem/storyofo.htm
  5. Iverson, Kenneth E., A Programming Language, Wiley, New York, 1962
    www.jsoftware.com/papers/APL.htm
  6. Iverson, Kenneth E., Elementary Functions: An Algorithmic Treatment, Science Research Associates, Inc., Chicago, 1966-03
    www.jsoftware.com/jwiki/Doc/Elementary_Functions_An_Algorithmic_Treatment
  7. Hui, Roger K.W., (ed.), APL Quotations and Anecdotes, 2010-09-18
    www.jsoftware.com/papers/APLQA.htm#APL_birthday
  8. Falkoff, Adin D. & Kenneth E. Iverson, The APL\360 Terminal System, Report RC-1922, IBM, 1967-10-16
    www.jsoftware.com/papers/APL360TerminalSystem.htm
  9. Conroy, C.A., Editor, APL\360 Newsletter Number 1, IBM, 1967-07
    bitsavers.informatik.uni-stuttgart.de/pdf/ibm/apl/APL_Newsletter_1_Jul67.pdf
  10. Falkoff, Adin D. & Kenneth E. Iverson, APL\360 User’s Manual, IBM,
    1968-08; Table 3.2 bitsavers.informatik.uni-stuttgart.de/pdf/ibm/apl/APL_360_Users_Manual_Aug68.pdf