November 23, 2005

Ab surd

Eugene Volokh writes:

Reading a book about the history of math, I came across the word surd. Never heard of it before, despite my many years of math education. I probably won't use it, precisely because if it's obscure to a fairly math-savvy person like me, it's probably obscure to others, too. But it's good to know, if only for Boggle purposes.

The AHD gives two definitions:

1. Mathematics An irrational number, such as √2. 2. Linguistics A voiceless sound in speech.

and a two-step etymology:

Medieval Latin surdus, speechless, surd (translation of Arabic (jaḏr) ’aṣamm, deaf (root), surd, translation of Greek alogos, speechless, surd), from Latin.

The OED explains the etymology as

[ad. L. surdus (in active sense) deaf, (in pass. sense) silent, mute, dumb, (of sound, etc.) dull, indistinct. The mathematical sense ‘irrational’ arises from L. surdus being used to render Gr. ἄλογος (Euclid bk. x. Def.), app. through the medium of Arab. açamm deaf, as in jaðr açamm surd root. ]

Pat Ballew's page "Origins of some arithmetic terms" attributes the following more extended account to Jeff Miller:

The Arabic translators in the ninth century translated the Greek rhetos (rational) by the Arabic muntaq (made to speak) and the Greek alogos (irrational) by the Arabic asamm (deaf, dumb). See e. g. W. Thomson, G. Junge, The Commentary of Pappus on Book X of Euclid's Elements, Cambridge: Harvard University Press, 1930 [Jan Hogendijk].

This was translated as surdus ("deaf" or "mute") in Latin.

As far as is known, the first known European to adopt this terminology was Gherardo of Cremona (c. 1150).

Fibonacci (1202) adopted the same term to refer to a number that has no root, according to Smith.

Surd is found in English in Robert Recorde's The Pathwaie to Knowledge (1551): "Quantitees partly rationall, and partly surde" (OED2).

According to Smith (vol. 2, page 252), there has never been a general agreement on what constitutes a surd. It is admitted that a number like sqrt 2 is a surd, but there have been prominent writers who have not included sqrt 6, since it is equal to sqrt 2 X sqrt 3. Smith also called the word surd "unnecessary and ill-defined" in his Teaching of Elementary Mathematics (1900).

G. Chrystal in Algebra, 2nd ed. (1889) says that "...a surd number is the incommensurable root of a commensurable number," and says that sqrt e is not a surd, nor is sqrt (1 + sqrt 2).

The extraordinary Liddell & Scott entry for logos explains that it is the verbal noun of legô (pick up; count, tell; say, speak) and gives this dizzying array of senses and sub-senses (with most cross-references, examples, citations and details edited out):

I. computation, reckoning
  1. account of money handled
    b. public accounts, i. e. branch of treasury
  2. generally, account, reckoning
  3. measure, tale
  4. esteem, consideration, value put on a person or thing
II. relation, correspondence, proportion
  1. generally
  2. Math., ratio, proportion
  3. Gramm., analogy, rule
III. explanation
  1. plea, pretext, ground
    b. plea, case, in Law or argument
    c. in Logic, proposition, whether as premiss or conclusion
    d. rule, principle, law, as embodying the result of logismos
  3. law, rule of conduct
  4. thesis, hypothesis, provisional ground
  5. reason, ground
  6. formula (wider than definition, but freq. equivalent thereto), term expressing reason
  7. reason, law exhibited in the world-process
    b. spermatikos l. generative principle in organisms
    c. in Neo-Platonic Philos., of regulative and formative forces, derived from the intelligible and operative in the sensible universe
IV. inward debate of the soul
  1. thinking, reasoning
  2. reason as a faculty
    b. creative reason
V. continuous statement, narrative (whether fact or fiction), oration, etc.
  1. fable
  2. legend
  3. tale, story
  4. speech, delivered in court, assembly, etc.
VI.verbal expression or utterance ...,rarely a single word, ... never in Gramm. signf. of vocable ..., usu. of a phrase
    a. pl., without Art.,talk
    b. sg., expression, phrase
    c. coupled or contrasted with words expressed or understood signifying act, fact, truth, etc., mostly in a depreciatory sense
    2.common talk, report, tradition
    c.mention, notice, description
    d. the talk one occasions, repute, mostly in good sense, good report, praise, honour ... less freq. in bad sense, evil report
    3. discussion, debate, deliberation
    b. right of discussion or speech
    c. dialogue, as a form of philosophical debate ... hence, dialogue as a form of literature
    d. section, division of a dialogue or treatise ... branch, department, division of a system of philosophy
    e. in pl., literature, letters ... but, also in pl., treatises
VII. a particular utterance, saying
  1. divine utterance, oracle
  2. proverb, maxim, saying
  3. assertion, opp. oath
  4. express resolution
  5. word of command, behest
VIII. thing spoken of, subject-matter
  2. plot of a narrative or dramatic poem
    b. in Art, subject of a painting
  3. thing talked of, event
IX. expression, utterance, speech regarded formally
  2. of various modes of expression, esp. artistic and literary
    b. of the constituents of lyric or dramatic poetry, words ...; dramatic dialogue
  3. Gramm., phrase, complex term ... l. onomatôdês noun-phrase
    b. sentence, complete statement
    c. language
X. the Word or Wisdom of God, personified as his agent in creation and world-government

So apparently the Arabic translators of Greek mathematical texts took alogos in L&S sense II.2, meaning "not a ratio (of integers)", and re-interpreted it as (an extended sense of) something like "not speaking", by reference to the L&S senses VI to IX of logos. (I'm sure they understood the concept of irrationality, the point at issue is just what they took the base sense of alogos to be, for the purpose of creating a calque.) Then [I falsely guessed -- ed.] they connected that to the idea of "not vocalized", as in the Arabic orthographic distinction between "vocalized" text in which short vowels are written, and "unvocalized" (i.e. normal) text in which they are not. (Was the traditional Arabic terminology really the same for rational vs. irrational numbers and vocalized vs. unvocalized text? I don't know... [But Ben Zimmer did -- see below]) Then when the Italians translated Arabic mathematics, they apparently used a literal translation of the Arabic term aṣamm for the irrational side of the opposition, but reverted to a lexical derivation of Latin ratio for the rational side.

The linguistic term surd is not used any more, at least not in any literatures that I read. For me, it evokes the days when phonetics involved instruments of glass, wood, leather, ivory and brass.

For the punchline, we go back to the Volokh post, where jc comments that

the link provides a good antonym: "that's not a surd, that's a sonant!" (that's for the other definition of surd - i'm not sure i understand what a voiceless sound in speech would be, though - if one's communication were reduced to such voiceless sounds, however, would it be a reductio ad surdum?)

[via email from Linda Seebach]

[Though it spoils the joke to explain the terminology, voiceless speech is of course speech during which the vocal cords are not vibrating; and a voiceless sound is one like [s] or [k] where this condition normally obtains.]

[Update: Karen Davis writes:

I first ran across this word in high school, in Cordwainer Smith (pen name of Paul Linebarger)'s short story "Alpha Ralpha Boulevard", in a quatrain apparently original with him:

She wasn't the woman I went to seek;
I met her by the merest chance.
She did not speak the French of France,
But the surded French of Martinique.

LION is ignorant of any works containing "surded", French or otherwise, but surd does turns up Erasmus Darwin's 1799 Botanic Garden, which includes in Part II ("Containing the Loves of Plants"), Canto II, in a sort of ode to the invented goddess Papyra:

119 ---Three favour'd youths her soft attention share,
120 The fond disciples of the studious Fair,
121 Hear her sweet voice, the golden process prove;
122 Gaze, as they learn; and, as they listen, love.
123 The first from Alpha to Omega joins
124 The letter'd tribes along the level lines;
125 Weighs with nice ear the vowel, liquid, surd,
126 And breaks in syllables the volant word.

Papyra's other two disciplines inscribe mathematics ("in deepening ranks his dexterous cypher-train") and music ("on four [sic] concordant lines / prints the lone crotchet, and the quaver joins").

And Delmore Schwartz used surd several times, always aspiring to the same Euclidian metaphor as in this passage about Coriolanus (from the seasonally appropriate work Act Four: "A Goodly House, the Feast Smells Well"):

176 And now the Volscian camp. As he shakes Rome,
177 He gnaws Aufidius with every tooth,
178 Unknowingly, though vowed to show, in all,
179 Humility, and move with modesty,
180 And loyalty.
181                               But the true surd
182 Is irreducible. The individual
183 Is uncontrollable. To him, to him
184 The soldiers draw, forget Aufidius,
185 Render him virtual kingship.

But it seems that Schwartz has forgotten his lessons, because the true surd is not an irreducible ratio, i.e. a ratio of mutually prime integers, but rather a number that can't be expressed as a ratio of integers in the first place.]

[Ben Zimmer corrects my guess that the Arabic deaf/mute vs. speaking metaphor might have been connected to the terminology of orthographic vocalization:

As far as I know, Arabic doesn't apply the deaf-mute vs. speaking metaphor to the voweling of texts as you suggest. Vowel markings are called Haraka:t, from the root حرك , and the act of voweling is denoted by a verb from the same root, literally meaning 'to set in motion'.

Now that I check the Hans Wehr dictionary, I see that the word for 'deaf' (aSamm or أصم) does actually have a linguistic as well as a mathematical sense, but it has nothing to do with voweling. The word can refer to a geminate verb, i.e., a triliteral verb where the second and third radicals are the same — also called mediae geminatae.


[ John Cowan sent in this quatrain by Lewis Carroll:

And what are all such gaieties to me,
   Whose thoughts are full of indices and surds?
x2 + 7x + 53
    = 11/3

This is one of those rare poems that includes a metrically scanned equation. The only other one that comes to mind at the moment is the limerick

    . #     .   .    #    .   .  #
    I used to think math was no fun
      .    .  #   .    .   #  .   .   #
    'Cause I couldn't see how it was done
     .  #  .     .  #
    Now Euler's my hero
     .  .  #   .    .  #
    For I now see why zero
    .  #  .   .  # .   .  #
    Is e to the pi i plus 1.

Caroll's quatrain includes a quadratic iambic pentameter (perhaps the only one so far explicitly composed as such?):

  .    #      .   # .  #   .   #  .    #
  x squared plus seven x plus fifty three

and then a trimeter with an inverted initial foot:

.  #   . # .    #
equals eleven thirds

Since Dodgson was a mathematician who loved puzzles, I wonder if there's a message in there? A bit of search on LION reveals that the quotation comes the first of Four Riddles, and Carroll's note says that

No. I. was written at the request of some young friends, who had gone to a ball at an Oxford Commemoration---and also as a specimen of what might be done by making the Double Acrostic a connected poem instead of what it has hitherto been, a string of disjointed stanzas, on every conceivable subject, and about as interesting to read straight through as a page of a Cyclopedia. The first two stanzas describe the two main words, and each subsequent stanza one of the cross "lights."

The whole poem is a bit long for this post, so you can find it here.

Maybe someone (John?) can tell me what word the equation is a clue to, or more precisely why Carroll chose that particular equation, among the large number that would have scanned and rhymed equally well.

A bit of high-school algebra yields

x2 + 7x + 148/3 = 0

x = (-7 ± sqrt(72 - 4*148/3))/2 = (-7 ± sqrt(49 - 592/3))/2 = (-7 ± sqrt(-148 1/3))/2

And since sqrt(-148 1/3) will be i * sqrt(148 1/3), there's a glimmer of something about "I surd one..." -- but maybe it doesn't mean anything after all, at least not gone at that way.


[I thought I was done with surd, but here is another update.

Trevor ap Patnarthur observed by email that "most Brits who went to downtown schools will be aware that a Surd is in fact a Sikh who wears the full rig. Hence Surd jokes."

This is apparently a re-spelling of Sardar -- Google indexes 54,100 {"Sardar jokes"} pages against only 634 for {"Surd jokes"}. The OED thinks that Sardar is a version of Sirdar, from [Urdū (Pers.) sardār, f. Pers. sar head + dār possessor.], meaning "in India and other Eastern countries, a military chief, a leader or general of a force or army". Hobson Jobson has an entry for sirdar, "s. Hind. from Pers. sar- dar, and less correctly sirdar, 'leader, a commander, an officer'; a chief, or lord; the head of a set of palankinbearers, and hence the 'sirdar-bearer,' or elliptically 'the Sirdar,' is in Bengal the style of the valet or body-servant, even when he may have no others under him."However, there are no pages of "Sirdar jokes", spelled as such.

I surmise that Sardar became a term for Sikhs because of their traditional role in the Indian army. The honorific form Sardarji is also apparently used -- there are 45,100 pages of {"Sardarji jokes"}.

The vowel change from Sardar to Surd is analogous to the change from pandit to pundit -- it reflects the Indic pronunciation of short /a/. The reason for the loss of the -ar is less clear to me.

Under whatever spelling, Surd/Sardar jokes seem to be a variant of Newfie/Polish/Chelm/Blonde/Aggie/men jokes:

Q: Why do men like surd jokes??
A: Because they can understand them.

OK, back to mashing the sweet potatoes. ]

Posted by Mark Liberman at November 23, 2005 05:24 AM