KEYNES FOLLOWS EDGEWORTH AMONG
THE DACTYLS
Geoffrey Fishburn
History of Economic Thought Society of Australia
27th Conference, Auckland, N.Z.
July 2014
School of Social Sciences
University of New South Wales
Sydney 2052
Australia.
[email protected]
1
The dactyl, a three-syllabic measure (‘foot’) of the form long-short-short, was a staple
of Classical (Greek and Roman) poetry, and gives its name to the dactylic hexameter, a line
of six feet where the dactyl appears a variable number of times (with its place otherwise most
usually being taken by the long-long spondee). Until the end of the Edwardian era, in Britain
at least, such a verse form was familiar to economists from their schooldays. Alfred Marshall,
for example, wrote of “… a Greek poet, whose hexameters may be rendered thus:…”
(Marshall 1919, 790 n1), the last of four attempts by him at this particular verse over a period
of some 16 years (Fishburn 2012). But one with far greater skill as a classicist was his
contemporary Francis Ysidro Edgeworth, most often known to economists for his seminal
contributions in economics and statistics as well as his role as editor of The Economic
Journal, but less often appreciated, and certainly largely glossed over in the familiar
biographical/obituary notes (Creedy 2008; Keynes 1926; Newman 2004; Price 1926), that
this was not the career path for which his undergraduate studies had prepared him, nor on
which he had first ventured.
Edgeworth’s application in 1875 for a Professorship in Greek at Bedford College,
London, was no less than would have been expected of one with a record of achievement
such as his in Classics at Trinity College, Dublin, and Oxford. The application, however, was
unsuccessful. It would be a few years before his neighbour in Hampstead, W.S. Jevons,
would show him the mathematical tools which could be used to fashion a more rigorous
version of Political Economy, a subject in which his interest had been stimulated whilst at
Oxford by Benjamin Jowett; but in the meantime he first produced, in 1877, his New and Old
Methods of Ethics in which, as Maynard Keynes later remarked:
Quotations from the Greek tread on the heels of the Differential Calculus,
and the philistine reader can scarcely tell whether it is a line of Homer or a
mathematical abstraction which is in course of integration. (Keynes 1926,
145) 1
It was only natural, then, for Edgeworth, when his mind in time turned to Statistics, to look to
the Classics, inter alia, for data. In the last week of June 1885, at the Jubilee Conference of
The Statistical Society of London he read a paper in which, as recorded in Nature of that
week:
A pretty illustration of important principles was afforded by the statistics of
a wasp’s nest … As further illustrations of the variety of interests amenable
to the general law, he adduced the attendance of the members of a club at a
table d’hôte, and the frequency of dactyls in the Latin hexameter. (Anon
1885, 188)
From the published paper in the Society’s Journal we read his transitions between these
“miscellaneous” subjects: first, “Ex.4. Returning to human society, we shall find that the
attendance at a club of its members fluctuates about as much as the coming in and going out
of the wasps”, and later, “Ex.5. From London clubs to Latin poetry is a violent transition;
significant of the variety of interests which are amenable to the Law of Error.” (Edgeworth
1885a, 210, 211) The data obtained from the Æneid he then uses first to characterise (in a
statistical sense) Virgil’s style, and then to propose a distinguishing test:
2
The number of dactyls in the hexameter affords a better example of the
mathematical method than might have been expected. I extracted the
modulus and Mean for ninety lines, which seemed a fair specimen of the
“Æneid,” those at the beginning of the fourth book. The Mean is 1.6,
excliusive of the fifth foot; the fluctuation is 1.6 also. We have here a test
of the Virgilian style. A passage of N lines is not Virgilian, if the mean
differ from 1.6 by two or three times the modulus 2
�1.6 �
1
90
1
+ 𝑁𝑁 � ;
provided that the opening lines of the fourth book may be taken as a sample
specimen of Virgilian style. (Edgeworth 1885a, 211. In the original a
1
typographical error gives the first term within brackets as 09.)
Edgeworth then applies this test to a more substantial extract from Virgil. The table of
observations is for convenience omitted here; but attention is drawn to evidence that
Edgeworth was no mere statistical bean-counter but was aware of the nuances of his database and, as a classicist, well able to appreciate such:
To determine this question I observed 600 3 additional hexameters
taken from the sixth book. The following table gives the results of the
whole set of Virgilian observations:-[…
…]
This table conveys many valuable lessons. First we learn that the test
above found is fairly accurate. It suffices to discriminate the Virgilian from
the Ovidian hexameter; which has an average of 2.2 and modulus 1.8.iv But
the test must not be pressed too far, or we should find Virgil himself
violating the canon of Virgilian style. … If we had taken the last thirty lines
of the passage under consideration, viz., “Æneid” VI, 434-464, we should
have found a still more marked exception to the general canon. Those thirty
lines pathetically describe the "plains of woe" where dwell unhappy lovers
:-“Lugentes campi, sic illos nomine dicunt.”
“The line too labours and the words move slow.” 5
The gusts of passion to which the rhythm is accommodated cannot be
treated as perfectly fortuitous. Nevertheless it appears that the causes at
work are sufficiently numerous and independent to fulfil the conditions
required for the elimination of chance. (Edgeworth 1885a, 211-212.)
Maynard Keynes did not read Classics at university, but was similarly a product of a school
system which, for those who could afford it, comprised little else. On 3 May, 1902 he read
before the Eton Literary Society (which he had revived) a paper on the long 12th century
poem De contemptu mundi (“Scorn for the World”) by the monk Bernard of Cluny. 6 On the
poem’s style he said:
3
The metre of the poem is probably better known than the poem itself; it is
technically known as ‘Leonini cristati trilices dactylici’; Bernard himself
refers to it as ‘tum dactylicum continuum exceptis finalibus, tum etiam
sonoritatem leonicam servans’. 7
It consists of couplets of pure dactylic hexameters with a two-syllable
rhyme at the end of the line, while in each single line the last two syllables
of the second foot rhyme with the last two of the fourth.
A year later, on 1 February 1903, he read before the Appenine Literary Society of King’s
College a paper on another 12th century figure, Peter Abelard. Describing the structure of
some hymns (Hymnarius Paraclitensis) written by Abelard for the convent which he had
founded, and of which Heloise was abbess, he says:
The first series (2 lines of 4 iambic feet followed by 2 of 5) . . .
and later:
After 9 cantos he changes to a dactylic metre
and finally:
In the next series he describes the infancy of Christ, beginning in an
awkward metre of 2 spondees followed by 2 dactyls: but he soon falls into a
more conventional form.
So, as we now know, Keynes could be relied on to know a dactyl when he saw one.
We next, after this commendable post-schoolboy effort, hear from Keynes on the
matter of dactyls in his A Treatise on Probability (1921).
In a characteristic passage Professor Edgeworth has applied these theories
to the frequency of dactyls in successive extracts from the Aeneid. The
mean for the line is 1.6, exclusive of the fifth foot, thus sharply
distinguishing the Virgilian line from the Ovidian, for which the
corresponding figure is 2.2.
…
That Edgeworth should have put forward this example in criticism of
Lexis’s conclusions, and that Lexis should have retorted that the
explanation was to be found in Edgeworth’s series not consisting of an
adequate number of observations, indicates, if I do not misapprehend them,
that these authorities are at fault in the principles, if not of probability, of
poetry.
The dactyls of the Virgilian hexameter are, in fact, a very good
example of what has been termed connexité, leading to subnormal
dispersion. The quantities of the successive feet are not independent, and
the appearance of a dactyl in one foot diminishes the probability of another
dactyl in that line. It is like the case of drawing black and white balls out of
an urn, where the balls are not replaced. (Keynes 1921, 437, 438. Italic
original.)
4
This then leaves us with two questions: the validity of Keynes’ criticism of Edgeworth in a
statistical context; and the validity of his understanding of the practice of Latin poetic
composition. As a preliminary we begin with a discussion of a concept which touches on both
questions:
“connexité, The quantities of the successive feet are not independent, and
the appearance of a dactyl in one foot diminishes the probability of another
dactyl in that line.” 8
Keynes offers no evidence in support of this notion; but curiously, it is from Edgeworth
elsewhere that he could have found such support, for this is along the lines of what
Edgeworth establishes, not in the paper under consideration by Keynes, but in another read
later in the same year before Section F of the British Association for the Advancement of
Science, and like the earlier paper published in the Journal of the Statistical Society of
London (Edgeworth 1885b). Whether Keynes was aware of this later paper is not obvious; in
any event, it does not appear in the bibliography of the Treatise.
As in the earlier paper Edgeworth’s data source is Virgil’s Æneid, in this instance VI.
270-394 and XI.1-150 (portions of IV and VI having been previously used). However, unlike
as in the earlier paper where his interest was in the frequency of dactyls per line, in this case
he is interested in the frequency of dactyls per foot: lines are divided into subgroups of five
(“quintettes”) and the number of the five having a dactyl in the first, second, third, and fourth
foot respectively is recorded -- the means are, from the first set of observations 3.08, 2.5,
1.86, 1.26, and from the second, 2.8, 2.3, 1.9, 1.3 (Edgeworth 1885b, 634, 638). Thus, on the
point of connexité at least, Keynes would appear to be right although Edgeworth had already
expressed the matter differently:
This conclusion may be verified by one of the humbler methods of
statistics. Out of fifteen trials [the first of the two sets], so to speak, the first
foot is only five times less than the second. The preponderance of the first
over the third and fourth feet is even more marked. We have heard of the
"hexameter rising aloft like a silvery column.” 9 We now see that this
column tapers up to its capital-the fifth foot with its almost constant dactyl.
(Edgeworth 1885b, 637. Italic original.)
In the following year Edgeworth read before the British Association a paper, “The
Mathematics of Banking”. Curious though it might seem in such a context the paper
(Edgeworth 1885b) just cited is reprised:
The nice relations between chance and law may receive additional
illustration from the statistics of other human interest at least as
complicated as banking – poetry; or not poetry, but disjecti membra poetæ
[scattered bits of poetry] I should perhaps say, in reference to the Virgilian
statistics which I presented to this Society last year. Take at random from
the Æneid 100 feet exclusive of fifth and six (sic) feet. You might take a
foot from each one of 100 consecutive lines; thus, one of the first four feet
5
from the first line, one from the second, and so on. Which of the four feet
you are to take on each occasion must be determineda in some random
fashion.
Well then, the average number of dactyls in the 100 feet thus taken
will be 40. And the excess above that average which isb very unlikely to be
overstepped is 18. If we could conceivec any practical purpose depending
on the possible provision of dactyls in 100 feet, it would be safe enough to
act upon the theorem just enunciated.
Let us now take 200 feet in the same way as before 100. The excess
above the average which will now afford the same safety, as 12 before is
not now 2x12=24, as common sense would probably take for granted, but
√2 x 12 = 17. Which proposition the reader will please to remember is not
merely a deduction from theory, but also fairly well verified by actual
observation.
The proposition must however be stated carefully. Suppose we took a
foot not indiscriminately from any of the first four places, but always from
the first place. We should then have in 100 feet a much higher average of
dactyls; the first foot being more frequently a dactyl than the others. The
excess of the new average over the old will be about 14. Now this excess
will be doubled when we take 200 instead of 100 lines. What is not doubled
when we double the number of feet examined, is the is the chance-deviation
in excess of the average – whether the average of lines in general or for a
particular category. (Edgeworth 1888, 125-6. Italic original.)
[Edgeworth’s footnotes; original numbers replaced by letters.]
a. E.g., tossing two unlike coins, and if head head turns up, taking the first
foot, if head tail the second foot, if tail head the third foot and if tail tail
the fourth foot.
b. Say thousands to one against such an event.
c. As if, in printing Latin verses, with the “longs” and ”shorts” marked, it
were requisite to deal out to different printers the number of each sort of
type which might be required for 100 Virgilian hexameters.
Returning then to Keynes, it would appear that he was, on the evidence of Edgeworth’s own
data, on the right track so far as connexité goes, that “the appearance of a dactyl in one foot
diminishes the probability of another dactyl in that line”. (The reason for qualification here
being that we could not deduce from Edgeworth’s data that the appearance of a dactyl in, say,
the fourth foot would diminish the probability of a dactyl appearing in any of the first, second
and third feet, but only that if one appears in the first foot, the probability of appearance
subsequently is diminished.)
6
Finally, and although not the principal concern of this paper, we turn to consideration
of aspects of Edgeworth’s statistical analysis (for a general perspective the reader is referred
to Stigler 1986, 322-24; 1999, 103-4).
Consider the exercise with which Keynes (on grounds entirely different from those
now mentioned) took issue: the extraction of mean and modulus from a “sample” of 90 lines
taken from Virgil’s Æneid (Book IV), to be then compared with 690 lines from Book VI.
However, the “sample”, whilst being appropriately “small” (90/705 lines of Book IV), is not
randomized, as we would now require of any statistical testing procedure: it is, in
Edgeworth’s words, “a fair specimen”. And Edgeworth would have been thoroughly steeped
in the text, with the result that Edgeworth-the-classicist surely guided the hand of Edgeworththe-(would be)-statistician.. A similar remark must apply to the selection of the comparison
lines (690/901) from Book VI; although this represents 72.6% of that Book we are dealing
here with a block (albeit large) of sequential lines, and not a random sample.
These were still formative days in the development of modern statistical analysis: for
instance, to test whether two means do or do not come from the same distribution he
employed (as seen earlier) the technique of the modulus of comparison, a technique by no
means universally accepted; the next generation (although still within his lifetime) would
hear of the work of W.S. Gosset (“Student” 1908).
At the centre of this enquiry is Edgeworth-the-statistician for whom Classics was but
one of many sources of data. It seems only appropriate that it should conclude with his words
on what he was attempting to do:
Such is the apparatus afforded by the Calculus of Probabilities for the
elimination of chance. But it must not be supposed that this mechanism is
always available in its perfect form. We must often be content with much
rougher indications of the degree of difference between two means which
might have been expected and is nothing extraordinary, than that which is
afforded by the regular extraction of the modulus. Still it is useful to have
the ideal of proof before our eyes, even when we cannot realize it in
practice. This function of the Calculus of Probabilities -- to present an
unattainable ideal – resembles that which the mathematical theory of
Political Economy performs (Edgeworth 1885a, 194)
7
References
Anon 1885. The Jubilee of the Statistical Society, Nature, 32(817). 188.
Barbé, Lluis 2010. Francis Ysidro Edgeworth. A Portrait with Family and Friends. (trams.
Mary C. Black). Cheltenham, U.K. and Northhampton, M.A., U.S.A.: Edward Elgar.
Bowley, A.L. 1928, F.Y. Edgeworth’s Contributions to Mathematical Statistics. London:
Royal Statistical Society.
Creedy, John 2008. Edgeworth, Francis Ysidro (1845–1926). The New Palgrave Dictionary
of Economics. Second Edition. Eds. Steven N. Durlauf and Lawrence E. Blume.
London: Palgrave Macmillan.
Edgeworth, F.Y. 1885a. Methods of Statistics, Journal of the Statistical Society of London,
Jubilee Volume (June 22 – 24). 181-217.
…………………1885b. On Methods of Ascertaining Variations in the Rate of Births,
Deaths, and Marriages. Journal of the Statistical Society of London, 48(4). 628-649.
………………… 1888. The Mathematical Theory of Banking. Journal of the Riyal
Statistical Society, 51(1). 113-127.
……………….. 1922a. Review: [untitled] Journal of the Royal Statistical Society, 85(1).
107-113
……………....... 1922b. The Philosophy of Chance. Mind, 31, No. 123. 257-283.
Fishburn, Geoffrey. 2011. Marx, Marshall, and 'the good water-nymphs'. History of
Economics Review, 54. 144-151.
Keynes, John Maynard 1921 (1973). A Treatise on Probability. (Collected WritingsVol.
VIII). Cambridge: Macmillan and Cambridge University Press.
------------------------------ 1926. Francis Ysidro Edgeworth (1845-1926). The Economic
Journal, 36(141). 140-153.
Marshall. Alfred. 1919. Industry and Trade. London: Macmillan and Co. Limited.
Newman, Peter (ed.) 2003. F.Y. Edgeworth: Mathematical Psychics and Further
Papers on Political Economy. Oxgord: Oxford University Press.
8
………………........ 2004. Edgeworth, Francis Ysidro (1845–1926). Oxford
Dictionary of National Biography. Oxford: Oxford University Press.
Pepin, Ronald E. 1991. Scorn for the World: Bernard of Cluny's De contemptu mundi. The
Latin text with English translation and an introduction. East Lansing, Mich.:
Colleagues Press.
Price, L.L. [L. L. P.] 1926. Francis Ysidro Edgeworth. Journal of the Royal Statistical
Society, 89(2). 371-377.
Raven, D.S. 1965. Latin Metre. London: Faber and Faber.
Schiller, Friedrich. 1797. Das Distichon. Musen-Almanach für das Jahr 1797. Tübingen: J.G.
Cotta. S, 67.
Shedd, W.G.T. (ed.). 1884. The Complete Works of Samuel Taylor Coleridge with an
introductory essay upon his philosophical and theological opinions. Vol. VII. New
York: Harper & Brothrs.
Stigler, Stephen M. 1986. The History Of Statistics. The Measurement of Uncertainty before
1900. Cambridge, MA and London, England: Harvard University Press.
------------------------ 1999. Statistics On The Table. The History of Statistical Concepts and
Methods. Cambridge, MA and London, England: Harvard University Press.
"Student". 1908. "The probable error of a mean". Biometrika 6(1). 1–25.
1
Keynes was here either exceptionally careless, or too clever by half. We are able to test
Keynes’s assertion with reference to material provided by Leofranc Holford-Strevens
(Newman 2003, 250-4). Apart from some words which could be found in almost any Greek
writer we find, contrary to Keynes, that the words and phrases used by Edgeworth come from
Aristotle, 12; Plato, 10; Aeschylus, 1; and only one from Homer (and this is well-removed
from any mathematical expression!). Ironically, although this particular line, concerning the
failure of Oeneus to sacrifice to Artemis, is translated (Newman 2003, 253) it is not sourced.
This is now done here:
9
[ἄλλοι δὲ θεοὶ δαίνυνθ᾽ ἑκατόμβας,
οἴῃ δ᾽ οὐκ ἔρρεξε Διὸς κούρῃ μεγάλοιο.]
ἢ λάθετ᾽ ἢ οὐκ ἐνόησεν: ἀάσατο δὲ μέγα θυμῷ.
(Hom.Il. 9.535-7; see also Luc. Symp. 25)
[the other gods feasted on hecatombs, and it was to the daughter of great Zeus alone that he
offered not,] that he either forgot or did not think of her, and was grievously misled in his
spirit.
Edgeworth’s use of Greek and Latin was lifelong:
At first such devices as the liberal use in his very earliest writing of
untranslated phrases or even whole sentences in Greek or Latin are
distracting, but as they become rarer in his later work we come to enjoy
them, as we might some other eccentricity in an old friend. (Stigler 1999,
95)
At the end of his life he was one of the very few survivors of the tradition
of free quotation from the Classics on all occasions and in all contexts.
(Keynes 1926, 143)
Edgeworth was certainly fond of inserting Latin (and occasionally, Greek) into his writings;
but he was not always accurate. In the paper principally considered here we read:
How then are we to ascertain the modulus, according to which any quantity
fulfilling the law of error fluctuates. To this we have reduced our problem:
Hie labor, hoc opus, est. (Edgeworth 1885a, 188)
In the original (Verg. A. VI. 129) the word order is otherwise, and there is no second comma
(underlining in original and translation added here):
… facilis descensus Averni:
noctes atque dies patet atri ianua Ditis;
sed revocare gradium superasque evadere ad auras.
hoc opus, hic labor est.
[The descent to hell is easy: both night and day the doors stand open. But to retrace one’s
steps and overcome the slope so as to emerge into the fresh air – this is the task, this is the
labour.]
Edgeworth’s memory for an apt Latin quote might not however have matched his Greek, as
we see (with footnote and original footnote marking) for example here with the play, as in the
original, on the “urn” as both repository of ashes and a container for lots:
This coincidence need not surprise, when it is remembered that Professor
Lexis found in many cases that deaths at certain ages fluctuate as I have
10
said, and are distributed just like damnatory lots at random taken from an
urn.
"Omnium
"Movetur urnâ . . . Sors." †
The conceptions of mythology and statistics coincide,
† The omitted words, "serius ocius exitura" (almost inconsistent with the
notion of pure sortition), are not applicable.
(Edgeworth 1885a, 205.)
This appears to have been taken from:
omnium
versatur urna serius ocius
sors exitura et nos in aeternum
exilium inpositura cumbae.
(Horace, Odes II, 3, 25)
Everyone’s lot is shaken by the grave urn and quickly we are carried by force forever away
on the funeral-barge.
Let c denote the modulus and σ the standard deviation of a distribution; then in
contemporary notation the relation is defined as c2 = 2σ2. “… [the modulus] is of course the
standard deviation … multiplied by √2. Edgeworth gives the name fluctuation to the quantity
c2” (Bowley 1928, 87). Or to put the relationship another way: the fluctuation is twice the
variance. Edgeworth himself, in the spirit of the Darwinian age, says: “The modulus
determines the species of the genus probability-curve…” (Edgeworth 1885a, 185 n.*).
2
3
As the Table (see original) shows, 690, not 600, lines were in fact observed. (In a modulus
of comparison calculation following the table Edgeworth uses the correct figure:
�1.4 �
work.
1
90
1
+ 690 � .) Typographical errors (as see earlier) were not unknown in Edgeworth’s
iv
Edgeworth gives no source for these figures, whether from his own personal calculation or
elsewhere.
5
This is a comment by Edgeworth and not a translation of the preceding line, which is
“Woeful fields, hence the name they call them.” (Æneid VI, 441)
6
I am grateful to Rod O’Donnell for providing access for me to this and the following paper
by Keynes.
7
Keynes chopped up the original:
11
Id enin genus metri, tum dactilum continuum exceptis finalibus trocheo vel spondee, tum
etiam sonoritatem leonicam servans ob sui difficultatem jam pene ne dicam penitus obsolevit.
“For this kind of metre, preserving not only the continuous dactyl, with the final trochee or
spondee excepted, but even leonine melodiousness, on account of its difficulty, now has
become almost (lest I say entirely) obsolete.” (Pepin 1991, 8-9)
leonine verse: a kind of Latin verse much used in the Middle Ages, consisting of hexameters
or alternate hexameters and pentameters, in which the final word rhymes with that
immediately preceding the cæsural pause. [OED]
8
No evidence of the use of this word in a statistical context other than by Keynes here has
been found. Its general definition, for which as synonyms are given cohérence, concordance,
corrélation, correspondance, relation, is: Rapport étroit qui existe entre deux ou plusieurs
choses (Larousse).
9
Edgeworth is here recalling the couplet of Samuel Taylor Coleridge:
In the hexameter rises the fountain's silvery column,
In the pentameter aye falling in melody back. (Shedd 1884, 332)
which Coleridge had taken from the original by Friedrich Schiller:
Im Hexameter steigt des Springquells silberne Säule,
Im Pentameter drauf fällt sie melodisch herab. (Schiller 1797)
It should be noted however that this refers to the structure of the elegiac diptych and
not, as Edgeworth is dealing with, successive lines of dactylic hexameters.