﻿ Vector, the Journal of the British APL Association

# Current issue

Vol.26 No.4

## Volumes

British APL Association

Archive articles posted online on request: ask the archivist.

Volume 26, No.1

`⍟ ○*`

# 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
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

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