My favourite APL symbol

by Roger K.W. Hui

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