On the Division of Labour
241. We have already mentioned what may, perhaps, appear
paradoxical to some of our readers that the division of labour
can be applied with equal success to mental as to mechanical
operations, and that it ensures in both the same economy of time.
A short account of its practical application, in the most
extensive series of calculations ever executed, will offer an
interesting illustration of this fact, whilst at the same time it
will afford an occasion for shewing that the arrangements which
ought to regulate the interior economy of a manufactory, are
founded on principles of deeper root than may have been supposed,
and are capable of being usefully employed in preparing the road
to some of the sublimest investigations of the human mind.
242. In the midst of that excitement which accompanied the
Revolution of France and the succeeding wars, the ambition of the
nation, unexhausted by its fatal passion for military renown, was
at the same time directed to some of the nobler and more
permanent triumphs which mark the era of a people's greatness and
which receive the applause of posterity long after their
conquests have been wrested from them, or even when their
existence as a nation may be told only by the page of history.
Amongst their enterprises of science, the French Government was
desirous of producing a series of mathematical tables, to
facilitate the application of the decimal system which they had
so recently adopted. They directed, therefore, their
mathematicians to construct such tables, on the most extensive
scale. Their most distinguished philosophers, responding fully to
the call of their country, invented new methods for this
laborious task; and a work, completely answering the large
demands of the Government, was produced in a remarkably short
period of time. M. Prony, to whom the superintendence of this
great undertaking was confided, in speaking of its commencement,
observes: Je m'y livrai avec toute l'ardeur dont j'etois capable,
et je m'occupai d'abord du plan general de l'execution. Toutes
les conditions que j'avois a remplir necessitoient l'emploi d'un
grand nombre de calculateurs; et il me vint bientot a la pensee
d'appliquer a la connection de ces Tables la division du travail,
dont les Arts de Commerce tirent un parti si avantageux pour
reunir a la pernection de main-d'oeuvre l'economie de la depense
et du temps. The circumstance which gave rise to this singular
application of the principle of the division on labour is so
interesting, that no apology is necessary for introducing it from
a small pamphlet printed at Paris a few years since, when a
proposition was made by the English to the French Government,
that the two countries should print these tables at their joint
expense.
243. The origin of the idea is related in the following
extract:
C'est a un chapitre d'un ouvrage Anglais,(1*) justement
celebre, (I.) qu'est probablement due l'existence de l'ouvrage
dont le gouvernement Britannique veut faire jouir le monde
savant:
Voici l'anecdote: M. de Prony s'etait engage. avec les
comites de gouvernement. a composer pour la division centesimale
du cercle, des tables logarithmiques et trigonometriques, qui,
non seulement ne laissassent rien a desirer quant a l'exactitude,
mais qui formassent le monument de calcul 1e plus vaste et le
plus imposant qui eut jamais ete execute, ou meme concu. Les
logarithmes des nombres de 1 a 200.000 formaient a ce travail un
supplement necessaire et exige. Il fut aise a M. de Prony de
s'assurer que meme en s'associant trois ou quatre habiles
co-operateurs. La plus grande duree presumable de sa vie ne lui
sufirai pas pour remplir ses engagements. Il etait occupe de
cette facheuse pensee lorsque. Se trouvant devant la boutique
d'un marchand de livres. Il appercut la belle edition Anglaise de
Smith, donnee a Londres en 1776: il ouvrit le livre au hazard. et
tomba sur le premier chapitre, qui traite de la division du
travail, et ou la fabrication des epingles est citee pour
exemple. A peine avait-il parcouru les premieres pages, que, par
une espece d'inspiration. il concut l'expedient de mettre ses
logarithmes en manufacture comme les epingles. Il faisait en ce
moment, a l'ecole polytechnique, des lecons sur une partie
d'analyse liee a ce genre de travail, la methode des differences,
et ses applications a l'interpolation. Il alla passer quelques
jours a la campagne. et revint a Paris avec le plan de
fabrication. qui a ete suivi dans l'execution. Il rassembla deux
ateliers. qui faisai ent separement les memes calculs, et se
servaient de verification reciproque.(2*)
244. The ancient methods of computing tables were altogether
inapplicable to such a proceeding. M. Prony, therefore, wishing
to avail himself of all the talent of his country in devising new
methods, formed the first section of those who were to take part
in this enterprise out of five or six of the most eminent
mathematicians in France.
First section. The duty of this first section was to
investigate, amongst the various analytical expressions which
could be found for the same function, that which was most readily
adapted to simple numerical calculation by many individuals
employed at the same time. This section had little or nothing to
do with the actual numerical work. When its labours were
concluded, the formulae on the use of which it had decided, were
delivered to the second section.
Second section. This section consisted of seven or eight
persons of considerable acquaintance with mathematics: and their
duty was to convert into numbers the formulae put into their
hands by the first section an operation of great labour; and then
to deliver out these formulae to the members of the third
section, and receive from them the finished calculations. The
members of this second section had certain means of verifying the
calculations without the necessity of repeating, or even of
examining, the whole of the work done by the third section.
Third section. The members of this section, whose number
varied from sixty to eighty, received certain numbers from the
second section, and, using nothing more than simple addition and
subtraction, they returned to that section the tables in a
finished state. It is remarkable that nine-tenths of this class
had no knowledge of arithmetic beyond the two first rules which
they were thus called upon to exercise, and that these persons
were usually found more correct in their calculations, than those
who possessed a more extensive knowledge of the subject.
245. When it is stated that the tables thus computed occupy
seventeen large folio volumes, some idea may perhaps be formed of
the labour. From that part executed by the third class, which may
almost be termed mechanical, requiring the least knowledge and by
far the greatest exertions, the first class were entirely exempt.
Such labour can always be purchased at an easy rate. The duties
of the second class, although requiring considerable skill in
arithmetical operations, were yet in some measure relieved by the
higher interest naturally felt in those more difficult
operations. The exertions of the first class are not likely to
require, upon another occasion, so much skill and labour as they
did upon the first attempt to introduce such a method; but when
the completion of a calculating engine shall have produced a
substitute for the whole of the third section of computers, the
attention of analysts will naturally be directed to simplifying
its application, by a new discussion of the methods of converting
analytical formulae into numbers.
246. The proceeding of M. Prony, in this celebrated system of
calculation, much resembles that of a skilful person about to
construct a cotton or silk mill, or any similar establishment.
Having, by his own genius, or through the aid of his friends,
found that some improved machinery may be successfully applied to
his pursuit, he makes drawings of his plans of the machinery, and
may himself be considered as constituting the first section. He
next requires the assistance of operative engineers capable of
executing the machinery he has designed, some of whom should
understand the nature of the processes to be carried on; and
these constitute his second section. When a sufficient number of
machines have been made, a multitude of other persons, possessed
of a lower degree of skill, must be employed in using them; these
form the third section: but their work, and the just performance
of the machines, must be still superintended by the second class.
247. As the possibility of performing arithmetical
calculations by machinery may appear to non-mathematical readers
to be rather too large a postulate, and as it is connected with
the subject of the division of labour, I shall here endeavour, in
a few lines, to give some slight perception of the manner in
which this can be done--and thus to remove a small portion of
the veil which covers that apparent mystery.
248. That nearly all tables of numbers which follow any law,
however complicated, may be formed, to a greater or less extent,
solely by the proper arrangement of the successive addition and
subtraction of numbers befitting each table, is a general
principle which can be demonstrated to those only who are well
acquainted with mathematics; but the mind, even of the reader who
is but very slightly acquainted with that science, will readily
conceive that it is not impossible, by attending to the following
example.
The subjoined table is the beginning of one in very extensive
use, which has been printed and reprinted very frequently in many
countries, and is called a table of square numbers.
Terms of Table A Table B first Difference C second Difference
1 1
3
2 4 2
5
3 9 2
7
4 16 2
9
5 25 2
11
6 36 2
13
7 49
Any number in the table, column A, may be obtained, by
multiplying the number which expresses the distance of that term
from the commencement of the table by itself; thus, 25 is the
fifth term from the beginning of the table, and 5 multiplied by
itself, or by 5, is equal to 25. Let us now subtract each term of
this table from the next succeeding term, and place the results
in another column (B), which may be called first difference
column. If we again subtract each term of this first difference
from the succeeding term, we find the result is always the number
2, (column C); and that the same number will always recur in that
column, which may be called the second difference, will appear to
any person who takes the trouble to carry on the table a few
terms further. Now when once this is admitted, it is quite clear
that, provided the first term (1) of the table, the first term
(3) of the first differences, and the first term (2) of the
second or constant difference, are originally given, we can
continue the table of square numbers to any extent, merely by
addition: for the series of first differences may be formed by
repeatedly adding the constant difference (2) to (3) the first
number in column B, and we then have the series of numbers, 3, 5,
6, etc.: and again, by successively adding each of these to the
first number (1) of the table, we produce the square numbers.
249. Having thus, I hope, thrown some light upon the
theoretical part of the question, I shall endeavour to shew that
the mechanical execution of such an engine, as would produce this
series of numbers, is not so far removed from that of ordinary
machinery as might be conceived.(3*) Let the reader imagine three
clocks, placed on a table side by side, each having only one
hand, and each having a thousand divisions instead of twelve
hours marked on the face; and every time a string is pulled, let
them strike on a bell the numbers of the divisions to which their
hands point. Let him further suppose that two of the clocks, for
the sake of distinction called B and C, have some mechanism by
which the clock C advances the hand of the clock B one division,
for each stroke it makes upon its own bell: and let the clock B
by a similar contrivance advance the hand of the clock A one
division, for each stroke it makes on its own bell. With such an
arrangement, having set the hand of the clock A to the division
I, that of B to III, and that of C to II, let the reader imagine
the repeating parts of the clocks to be set in motion continually
in the following order: viz.--pull the string of clock A; pull
the string of clock B; pull the string of clock C.
The table on the following page will then express the series
of movements and their results.
If now only those divisions struck or pointed at by the clock
A be attended to and written down, it will be found that they
produce the series of the squares of the natural numbers. Such a
series could, of course, be carried by this mechanism only so far
as the numbers which can be expressed by three figures; but this
may be sufficient to give some idea of the construction--and
was, in fact, the point to which the first model of the
calculating engine, now in progress, extended.
250. We have seen, then, that the effect of the division of
labour, both in mechanical and in mental operations, is, that it
enables us to purchase and apply to each process precisely that
quantity of skill and knowledge which is required for it: we
avoid employing any part of the time of a man who can get eight
or ten shillings a day by his skill in tempering needles, in
turning a wheel, which can be done for sixpence a day; and we
equally avoid the loss arising from the employment of an
accomplished mathematician in performing the lowest processes of
arithmetic.
251. The division of labour cannot be successfully practised
unless there exists a great demand for its produce; and it
requires a large capital to be employed in those arts in which it
is used. In watchmaking it has been carried, perhaps, to the
greatest extent. It was stated in evidence before a committee of
the House of Commons, that there are a hundred and two distinct
branches of this art, to each of which a boy may be put
apprentice: and that he only learns his master's department, and
is unable, after his apprenticeship has expired, without
subsequent instruction, to work at any other branch. The
watch-finisher, whose business is to put together the scattered
parts, is the only one, out of the hundred and two persons, who
can work in any other department than his own.
252. In one of the most difficult arts, that of mining, great
improvements have resulted from the judicious distribution of the
duties; and under the arrangments which have gradually been
introduced, the whole system of the mine and its government is
now placed under the control of the following officers.
1. A manager, who has the general knowledge of all that is to
be done, and who may be assisted by one or more skilful persons.
2. Underground captains direct the proper mining operations,
and govern the working miners.
3. The purser and book-keeper manage the accounts.
4. The engineer erects the engines, and superintends the men
who work them.
5. A chief pitman has charge of the pumps and the apparatus
of the shafts.
6. A surface-captain, with assistants, receives the ores
raised, and directs the dressing department, the object of which
is to render them marketable.
7. The head carpenter superintends many constructions.
8. The foreman of the smiths regulates the ironwork and
tools.
9. A materials man selects, purchases, receives and delivers
all articles required.
10. The roper has charge of ropes and cordage of all sorts.
Notes:
1. An Enquiry into the Nature and Causes of the Wealth of
Nations, by Adam Smith.
2. Note sur la publication, proposee par le gouvernement Anglais
des grandes tables logarithmiques et trigonometriques de M de
Prony De l'imprimerie de F. Didot, December 1, 1829, p. 7
3. Since the publication of the second edition of this work, one
portion of the engine which I have been constructing for some
years past has been put together. It calculates, in three
columns, a table with its first and second differences. Each
column can be expressed as far as five figures, so that these
fifteen figures constitute about one ninth part of the larger
engine. The ease and precision with which it works leave no room
to doubt its success in the more extended form. Besides tables of
squares, cubes, and portions of logarithmic tables, it possesses
the power of calculating certain series whose differences are not
constant; and it has already tabulated parts of series formed
from the following equations:
The third differential of ux = units figur of delta ux
The third differential of ux = nearest whole no. to (1/10,000
delta ux)
The subjoined is one amongst the series which it has calculated:
0 3,486 42,972
0 4,991 50,532
1 6,907 58,813
14 9,295 67,826
70 12,236 77,602
230 15,741 88,202
495 19,861 99,627
916 24,597 111,928
1,504 30,010 125,116
2,340 36,131 139,272
The general term of this is,
ux = (x(x-1)(x-2))/(1 X 2 X 3) + the whole number in x/10 +
10 Sigma^3 (units figure of (x(x-1)/2)